id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-9600 | Primitive of Cosine of a x over Sine of a x minus Cosine of a x | :$\ds \int \frac {\cos a x \rd x} {\sin a x - \cos a x} = \frac {-x} 2 + \frac 1 {2 a} \ln \size {\sin a x - \cos a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\cos a x \rd x} {\sin a x - \cos a x}
| r = \int \frac {\paren {\cos a x - \sin a x + \sin a x} \rd x} {\sin a x - \cos a x}
| c =
}}
{{eqn | r = -\int \frac {\paren {\sin a x - \cos a x} \rd x} {\sin a x - \cos a x} + \int \frac {\sin a x \rd x} {\sin a x - \cos a x}
... | :$\ds \int \frac {\cos a x \rd x} {\sin a x - \cos a x} = \frac {-x} 2 + \frac 1 {2 a} \ln \size {\sin a x - \cos a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\cos a x \rd x} {\sin a x - \cos a x}
| r = \int \frac {\paren {\cos a x - \sin a x + \sin a x} \rd x} {\sin a x - \cos a x}
| c =
}}
{{eqn | r = -\int \frac {\paren {\sin a x - \cos a x} \rd x} {\sin a x - \cos a x} + \int \frac {\sin a x \rd x} {\sin a x - \cos a x}
... | Primitive of Cosine of a x over Sine of a x minus Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_Sine_of_a_x_minus_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_Sine_of_a_x_minus_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Sine of a x over Sine of a x minus Cosine of a x"
] |
proofwiki-9601 | Primitive of Sine of a x over p plus q of Cosine of a x | :$\ds \int \frac {\sin a x \rd x} {p + q \cos a x} = \frac {-1} {a q} \ln \size {p + q \cos a x} + C$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {p + q \cos a x}
| r = -a q \sin a x
| c = Derivative of $\cos a x$
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sin a x \rd x} {p + q \cos a x}
| r = \frac {-1} {a q} \ln \size {p + q \cos a x} + C
| c = Primitive of Function under its Deriv... | :$\ds \int \frac {\sin a x \rd x} {p + q \cos a x} = \frac {-1} {a q} \ln \size {p + q \cos a x} + C$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {p + q \cos a x}
| r = -a q \sin a x
| c = [[Derivative of Cosine of a x|Derivative of $\cos a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sin a x \rd x} {p + q \cos a x}
| r = \frac {-1} {a q} \ln \size {p + q \cos a x} + C
| c = [[Pri... | Primitive of Sine of a x over p plus q of Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_p_plus_q_of_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_p_plus_q_of_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Derivative of Cosine Function/Corollary",
"Primitive of Function under its Derivative"
] |
proofwiki-9602 | Primitive of Cosine of a x over p plus q of Sine of a x | :$\ds \int \frac {\cos a x \rd x} {p + q \sin a x} = \frac 1 {a q} \ln \size {p + q \sin a x} + C$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {p + q \sin a x}
| r = a q \cos a x
| c = Derivative of $\sin a x$
}}
{{eqn | ll= \leadsto
| l = \int \frac {\cos a x \rd x} {p + q \sin a x}
| r = \frac 1 {a q} \ln \size {p + q \sin a x} + C
| c = Primitive of Function under its Derivativ... | :$\ds \int \frac {\cos a x \rd x} {p + q \sin a x} = \frac 1 {a q} \ln \size {p + q \sin a x} + C$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {p + q \sin a x}
| r = a q \cos a x
| c = [[Derivative of Sine of a x|Derivative of $\sin a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\cos a x \rd x} {p + q \sin a x}
| r = \frac 1 {a q} \ln \size {p + q \sin a x} + C
| c = [[Primitive... | Primitive of Cosine of a x over p plus q of Sine of a x | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_p_plus_q_of_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_p_plus_q_of_Sine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Derivative of Sine Function/Corollary",
"Primitive of Function under its Derivative"
] |
proofwiki-9603 | Primitive of Sine of a x over Power of p plus q of Cosine of a x | :$\ds \int \frac {\sin a x \rd x} {\paren {p + q \cos a x}^n} = \frac 1 {a q \paren {n - 1} \paren {p + q \cos a x}^{n - 1} } + C$ | {{begin-eqn}}
{{eqn | l = z
| r = p + q \cos a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -a q \sin a x
| c = Derivative of $\cos a x$
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sin a x \rd x} {\paren {p + q \cos a x}^n}
| r = \int \frac {\d z} {-a q z^n}
... | :$\ds \int \frac {\sin a x \rd x} {\paren {p + q \cos a x}^n} = \frac 1 {a q \paren {n - 1} \paren {p + q \cos a x}^{n - 1} } + C$ | {{begin-eqn}}
{{eqn | l = z
| r = p + q \cos a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -a q \sin a x
| c = [[Derivative of Cosine of a x|Derivative of $\cos a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sin a x \rd x} {\paren {p + q \cos a x}^n}
| r... | Primitive of Sine of a x over Power of p plus q of Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_Power_of_p_plus_q_of_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_Power_of_p_plus_q_of_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Derivative of Cosine Function/Corollary",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9604 | Primitive of Cosine of a x over Power of p plus q of Sine of a x | :$\ds \int \frac {\cos a x \rd x} {\paren {p + q \sin a x}^n} = \frac {-1} {a q \paren {n - 1} \paren {p + q \sin a x}^{n - 1} } + C$ | {{begin-eqn}}
{{eqn | l = z
| r = p + q \sin a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a q \sin a x
| c = Derivative of $\sin a x$
}}
{{eqn | ll= \leadsto
| l = \int \frac {\cos a x \rd x} {\paren {p + q \sin a x}^n}
| r = \int \frac {\d z} {a q z^n}
... | :$\ds \int \frac {\cos a x \rd x} {\paren {p + q \sin a x}^n} = \frac {-1} {a q \paren {n - 1} \paren {p + q \sin a x}^{n - 1} } + C$ | {{begin-eqn}}
{{eqn | l = z
| r = p + q \sin a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a q \sin a x
| c = [[Derivative of Sine of a x|Derivative of $\sin a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\cos a x \rd x} {\paren {p + q \sin a x}^n}
| r = ... | Primitive of Cosine of a x over Power of p plus q of Sine of a x | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_Power_of_p_plus_q_of_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_Power_of_p_plus_q_of_Sine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Derivative of Sine Function/Corollary",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9605 | Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x | :$\ds \int \frac {\d x} {p \sin a x + q \cos a x} = \frac 1 {a \sqrt {p^2 + q^2} } \ln \tan \size {\frac {a x + \arctan \dfrac q p} 2} + C$ | === Lemma ===
{{:Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x/Lemma}}{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p \sin a x + q \cos a x}
| r = \int \frac {\d x} {\sqrt {p^2 + q^2} \map \cos {a x + \arctan \dfrac {-p} q} }
| c = Multiple of Sine plus Multiple of Cosine
... | :$\ds \int \frac {\d x} {p \sin a x + q \cos a x} = \frac 1 {a \sqrt {p^2 + q^2} } \ln \tan \size {\frac {a x + \arctan \dfrac q p} 2} + C$ | === [[Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x/Lemma|Lemma]] ===
{{:Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x/Lemma}}{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p \sin a x + q \cos a x}
| r = \int \frac {\d x} {\sqrt {p^2 + q^2} \map \cos {a x ... | Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_by_Sine_of_a_x_plus_q_by_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_by_Sine_of_a_x_plus_q_by_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x/Lemma",
"Multiple of Sine plus Multiple of Cosine",
"Secant is Reciprocal of Cosine",
"Primitive of Function of a x + b",
"Primitive of Secant of a x",
"Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x/Lemma"
] |
proofwiki-9606 | Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x plus r | :$\ds \int \frac {\d x} {p \sin a x + q \cos a x + r} = \begin{cases}
\ds \frac 2 {a \sqrt {r^2 - p^2 - q^2} } \map \arctan {\frac {p + \paren {r - q} \tan \dfrac {a x} 2} {\sqrt {r^2 - p^2 - q^2} } } + C & : p^2 + q^2 < r^2 \\
\ds \frac 1 {a \sqrt {p^2 + q^2 - r^2} } \ln \size {\frac {p - \sqrt {p^2 + q^2 - r^2} + \pa... | Let $u = \tan \dfrac {a x} 2$.
Then we have:
{{begin-eqn}}
{{eqn | l = \d x
| r = \dfrac {2 \rd u} {a \paren {1 + u^2} }
}}
{{eqn | l = \sin a x
| r = \dfrac {2 u} {1 + u^2}
}}
{{eqn | l = \cos a x
| r = \dfrac {1 - u^2} {1 + u^2}
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p \s... | :$\ds \int \frac {\d x} {p \sin a x + q \cos a x + r} = \begin{cases}
\ds \frac 2 {a \sqrt {r^2 - p^2 - q^2} } \map \arctan {\frac {p + \paren {r - q} \tan \dfrac {a x} 2} {\sqrt {r^2 - p^2 - q^2} } } + C & : p^2 + q^2 < r^2 \\
\ds \frac 1 {a \sqrt {p^2 + q^2 - r^2} } \ln \size {\frac {p - \sqrt {p^2 + q^2 - r^2} + \pa... | Let $u = \tan \dfrac {a x} 2$.
Then we have:
{{begin-eqn}}
{{eqn | l = \d x
| r = \dfrac {2 \rd u} {a \paren {1 + u^2} }
}}
{{eqn | l = \sin a x
| r = \dfrac {2 u} {1 + u^2}
}}
{{eqn | l = \cos a x
| r = \dfrac {1 - u^2} {1 + u^2}
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {... | Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x plus r | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_by_Sine_of_a_x_plus_q_by_Cosine_of_a_x_plus_r | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_by_Sine_of_a_x_plus_q_by_Cosine_of_a_x_plus_r | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Weierstrass Substitution",
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2"
] |
proofwiki-9607 | Primitive of Reciprocal of p by Sine of a x plus q by 1 plus Cosine of a x | :$\ds \int \frac {\rd x} {p \sin a x + q \paren {1 + \cos a x} } = \frac 1 {a p} \ln \size {q + p \tan \frac {a x} 2} + C$ | Let $z = a x$.
Then $\d x = \dfrac {\d z} a$ and so:
$(1): \quad \ds \int \frac {\rd x} {p \sin a x + q \paren {1 + \cos a x} } = \dfrac 1 a \int \frac {\rd z} {p \sin z + q \paren {1 + \cos z} }$
Then:
{{begin-eqn}}
{{eqn | l = u
| r = \tan \frac z 2
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\... | :$\ds \int \frac {\rd x} {p \sin a x + q \paren {1 + \cos a x} } = \frac 1 {a p} \ln \size {q + p \tan \frac {a x} 2} + C$ | Let $z = a x$.
Then $\d x = \dfrac {\d z} a$ and so:
$(1): \quad \ds \int \frac {\rd x} {p \sin a x + q \paren {1 + \cos a x} } = \dfrac 1 a \int \frac {\rd z} {p \sin z + q \paren {1 + \cos z} }$
Then:
{{begin-eqn}}
{{eqn | l = u
| r = \tan \frac z 2
| c =
}}
{{eqn | ll= \leadsto
| l = \int \fr... | Primitive of Reciprocal of p by Sine of a x plus q by 1 plus Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_by_Sine_of_a_x_plus_q_by_1_plus_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_by_Sine_of_a_x_plus_q_by_1_plus_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Weierstrass Substitution",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Primitive of Reciprocal of a x + b"
] |
proofwiki-9608 | Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x plus Root of p squared plus q squared | :$\ds \int \frac {\d x} {p \sin a x + q \cos a x + \sqrt {p^2 + q^2} } = \frac {-1} {a \sqrt {p^2 + q^2} } \map \tan {\frac \pi 4 - \frac {a x + \arctan \frac q p} 2} + C$ | Let $\theta = \arctan \dfrac p q$.
Then by the definitions of sine, cosine and tangent:
{{begin-eqn}}
{{eqn | n = 1
| l = \cos \theta
| r = \frac q {\sqrt {p^2 + q^2} }
}}
{{eqn | l = \sin \theta
| r = \frac p {\sqrt {p^2 + q^2} }
}}
{{end-eqn}}
Now consider:
{{begin-eqn}}
{{eqn | l = \map \cos {\frac... | :$\ds \int \frac {\d x} {p \sin a x + q \cos a x + \sqrt {p^2 + q^2} } = \frac {-1} {a \sqrt {p^2 + q^2} } \map \tan {\frac \pi 4 - \frac {a x + \arctan \frac q p} 2} + C$ | Let $\theta = \arctan \dfrac p q$.
Then by the definitions of [[Definition:Sine of Angle|sine]], [[Definition:Cosine of Angle|cosine]] and [[Definition:Tangent of Angle|tangent]]:
{{begin-eqn}}
{{eqn | n = 1
| l = \cos \theta
| r = \frac q {\sqrt {p^2 + q^2} }
}}
{{eqn | l = \sin \theta
| r = \frac ... | Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x plus Root of p squared plus q squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_by_Sine_of_a_x_plus_q_by_Cosine_of_a_x_plus_Root_of_p_squared_plus_q_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_by_Sine_of_a_x_plus_q_by_Cosine_of_a_x_plus_Root_of_p_squared_plus_q_squared | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Definition:Sine/Definition from Triangle",
"Definition:Cosine/Definition from Triangle",
"Definition:Tangent Function/Definition from Triangle",
"Cosine of Difference",
"Sine of Complement equals Cosine",
"Sine of Sum",
"Arctangent of Reciprocal equals Arccotangent",
"Tangent of Complement equals Cot... |
proofwiki-9609 | Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x minus Root of p squared plus q squared | :$\ds \int \frac {\d x} {p \sin a x + q \cos a x - \sqrt {p^2 + q^2} } = \frac {-1} {a \sqrt {p^2 + q^2} } \map \tan {\frac \pi 4 + \frac {a x + \arctan \frac q p} 2} + C$ | Let $\theta = \arctan \dfrac p q$.
Then by the definitions of sine, cosine and tangent:
{{begin-eqn}}
{{eqn | n = 1
| l = \cos \theta
| r = \frac q {\sqrt {p^2 + q^2} }
}}
{{eqn | l = \sin \theta
| r = \frac p {\sqrt {p^2 + q^2} }
}}
{{end-eqn}}
Now consider:
{{begin-eqn}}
{{eqn | l = \map \cos {\frac... | :$\ds \int \frac {\d x} {p \sin a x + q \cos a x - \sqrt {p^2 + q^2} } = \frac {-1} {a \sqrt {p^2 + q^2} } \map \tan {\frac \pi 4 + \frac {a x + \arctan \frac q p} 2} + C$ | Let $\theta = \arctan \dfrac p q$.
Then by the definitions of [[Definition:Sine of Angle|sine]], [[Definition:Cosine of Angle|cosine]] and [[Definition:Tangent of Angle|tangent]]:
{{begin-eqn}}
{{eqn | n = 1
| l = \cos \theta
| r = \frac q {\sqrt {p^2 + q^2} }
}}
{{eqn | l = \sin \theta
| r = \frac ... | Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x minus Root of p squared plus q squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_by_Sine_of_a_x_plus_q_by_Cosine_of_a_x_minus_Root_of_p_squared_plus_q_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_by_Sine_of_a_x_plus_q_by_Cosine_of_a_x_minus_Root_of_p_squared_plus_q_squared | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Definition:Sine/Definition from Triangle",
"Definition:Cosine/Definition from Triangle",
"Definition:Tangent Function/Definition from Triangle",
"Cosine of Difference",
"Sine of Complement equals Cosine",
"Sine of Sum",
"Arctangent of Reciprocal equals Arccotangent",
"Tangent of Complement equals Cot... |
proofwiki-9610 | Primitive of Reciprocal of p squared by square of Sine of a x plus q squared by square of Cosine of a x | :$\ds \int \frac {\d x} {p^2 \sin^2 a x + q^2 \cos^2 a x} = \frac 1 {a p q} \map \arctan {\frac {p \tan a x} q} + C$ | Let $u = p^2 + q^2$ and $v = q^2 - p^2$.
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = u + v
| r = 2 q^2
}}
{{eqn | n = 2
| l = u - v
| r = 2 p ^2
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = u^2 - v^2
| r = \paren {u + v} \paren {u - v}
}}
{{eqn | l = u^2 - v^2
| r = \paren {2 q^2} \par... | :$\ds \int \frac {\d x} {p^2 \sin^2 a x + q^2 \cos^2 a x} = \frac 1 {a p q} \map \arctan {\frac {p \tan a x} q} + C$ | Let $u = p^2 + q^2$ and $v = q^2 - p^2$.
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = u + v
| r = 2 q^2
}}
{{eqn | n = 2
| l = u - v
| r = 2 p ^2
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = u^2 - v^2
| r = \paren {u + v} \paren {u - v}
}}
{{eqn | l = u^2 - v^2
| r = \paren {2 q^2... | Primitive of Reciprocal of p squared by square of Sine of a x plus q squared by square of Cosine of a x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_by_square_of_Sine_of_a_x_plus_q_squared_by_square_of_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_by_square_of_Sine_of_a_x_plus_q_squared_by_square_of_Cosine_of_a_x/Proof_1 | [
"Primitives involving Sine Function and Cosine Function",
"Primitive of Reciprocal of p squared by square of Sine of a x plus q squared by square of Cosine of a x"
] | [] | [
"Power Reduction Formulas/Sine Squared",
"Power Reduction Formulas/Cosine Squared",
"Primitive of Reciprocal of p plus q by Cosine of a x"
] |
proofwiki-9611 | Primitive of Reciprocal of p squared by square of Sine of a x plus q squared by square of Cosine of a x | :$\ds \int \frac {\d x} {p^2 \sin^2 a x + q^2 \cos^2 a x} = \frac 1 {a p q} \map \arctan {\frac {p \tan a x} q} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 \sin^2 a x + q^2 \cos^2 a x}
| r = \int \frac {\sec^2 a x \d x} {p^2 \tan^2 a x + q^2}
| c = multiplying by $\dfrac {\sec^2 a x} {\sec^2 a x}$
}}
{{eqn | r = \frac 1 a \int \frac {\d t} {p^2 t^2 + q^2}
| c = substituting $t = \tan a x$
}}
{{eqn | r = \frac 1 {a p^2} \... | :$\ds \int \frac {\d x} {p^2 \sin^2 a x + q^2 \cos^2 a x} = \frac 1 {a p q} \map \arctan {\frac {p \tan a x} q} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 \sin^2 a x + q^2 \cos^2 a x}
| r = \int \frac {\sec^2 a x \d x} {p^2 \tan^2 a x + q^2}
| c = multiplying by $\dfrac {\sec^2 a x} {\sec^2 a x}$
}}
{{eqn | r = \frac 1 a \int \frac {\d t} {p^2 t^2 + q^2}
| c = [[Integration by Substitution|substituting]] $t = \tan a x$
... | Primitive of Reciprocal of p squared by square of Sine of a x plus q squared by square of Cosine of a x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_by_square_of_Sine_of_a_x_plus_q_squared_by_square_of_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_by_square_of_Sine_of_a_x_plus_q_squared_by_square_of_Cosine_of_a_x/Proof_2 | [
"Primitives involving Sine Function and Cosine Function",
"Primitive of Reciprocal of p squared by square of Sine of a x plus q squared by square of Cosine of a x"
] | [] | [
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-9612 | Primitive of Reciprocal of p squared by square of Sine of a x minus q squared by square of Cosine of a x | :$\ds \int \frac {\d x} {p^2 \sin^2 a x - q^2 \cos^2 a x} = \frac 1 {2 a p q} \ln \size {\frac {p \tan a x - q} {p \tan a x + q} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 \sin^2 a x - q^2 \cos^2 a x}
| r = \int \frac {\sec^2 a x} {p^2 \tan^2 a x - q^2} \rd x
| c = multiplying by $\dfrac {\sec^2 a x} {\sec^2 a x}$
}}
{{eqn | r = \frac 1 a \int \frac 1 {p^2 t^2 - q^2} \rd t
| c = substituting $t = \tan a x$
}}
{{eqn | r = ... | :$\ds \int \frac {\d x} {p^2 \sin^2 a x - q^2 \cos^2 a x} = \frac 1 {2 a p q} \ln \size {\frac {p \tan a x - q} {p \tan a x + q} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 \sin^2 a x - q^2 \cos^2 a x}
| r = \int \frac {\sec^2 a x} {p^2 \tan^2 a x - q^2} \rd x
| c = multiplying by $\dfrac {\sec^2 a x} {\sec^2 a x}$
}}
{{eqn | r = \frac 1 a \int \frac 1 {p^2 t^2 - q^2} \rd t
| c = [[Integration by Substitution|substituting]... | Primitive of Reciprocal of p squared by square of Sine of a x minus q squared by square of Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_by_square_of_Sine_of_a_x_minus_q_squared_by_square_of_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_by_square_of_Sine_of_a_x_minus_q_squared_by_square_of_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Integration by Substitution",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form"
] |
proofwiki-9613 | Primitive of Power of Sine of a x by Power of Cosine of a x/Reduction of Power of Sine | :$\ds \int \sin^m a x \cos^n a x \rd x = \frac {-\sin^{m - 1} a x \cos^{n + 1} a x} {a \paren {m + n} } + \frac {m - 1} {m + n} \int \sin^{m - 2} a x \cos^n a x \rd x + C$
for $n \ne -m$. | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sin^{m - 1} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \paren {m - 1} \sin^{m - 2} a x \cos a x
... | :$\ds \int \sin^m a x \cos^n a x \rd x = \frac {-\sin^{m - 1} a x \cos^{n + 1} a x} {a \paren {m + n} } + \frac {m - 1} {m + n} \int \sin^{m - 2} a x \cos^n a x \rd x + C$
for $n \ne -m$. | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sin^{m - 1} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \paren {m - 1} ... | Primitive of Power of Sine of a x by Power of Cosine of a x/Reduction of Power of Sine | https://proofwiki.org/wiki/Primitive_of_Power_of_Sine_of_a_x_by_Power_of_Cosine_of_a_x/Reduction_of_Power_of_Sine | https://proofwiki.org/wiki/Primitive_of_Power_of_Sine_of_a_x_by_Power_of_Cosine_of_a_x/Reduction_of_Power_of_Sine | [
"Primitive of Power of Sine of a x by Power of Cosine of a x"
] | [] | [
"Definition:Primitive",
"Derivative of Sine Function/Corollary",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Primitive of Power of Cosine of a x by Sine of a x",
"Integration by Parts",
"Sum of Squares of Sine and Cosine",
"Linear Combination of Integrals/Indefinite",
"Definit... |
proofwiki-9614 | Primitive of Power of Sine of a x by Power of Cosine of a x/Reduction of Power of Cosine | :$\ds \int \sin^m a x \cos^n a x \rd x = \frac {\sin^{m + 1} a x \cos^{n - 1} a x} {a \paren {m + n} } + \frac {n - 1} {m + n} \int \sin^m a x \cos^{n - 2} a x \rd x + C$
for $n \ne -m$. | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \cos^{n - 1} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -a \paren {n - 1} \cos^{n - 2} a x \sin a x... | :$\ds \int \sin^m a x \cos^n a x \rd x = \frac {\sin^{m + 1} a x \cos^{n - 1} a x} {a \paren {m + n} } + \frac {n - 1} {m + n} \int \sin^m a x \cos^{n - 2} a x \rd x + C$
for $n \ne -m$. | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \cos^{n - 1} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -a \paren {n - 1}... | Primitive of Power of Sine of a x by Power of Cosine of a x/Reduction of Power of Cosine | https://proofwiki.org/wiki/Primitive_of_Power_of_Sine_of_a_x_by_Power_of_Cosine_of_a_x/Reduction_of_Power_of_Cosine | https://proofwiki.org/wiki/Primitive_of_Power_of_Sine_of_a_x_by_Power_of_Cosine_of_a_x/Reduction_of_Power_of_Cosine | [
"Primitive of Power of Sine of a x by Power of Cosine of a x"
] | [] | [
"Definition:Primitive",
"Derivative of Cosine Function/Corollary",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Primitive of Power of Sine of a x by Cosine of a x",
"Integration by Parts",
"Sum of Squares of Sine and Cosine",
"Linear Combination of Integrals/Indefinite",
"Defin... |
proofwiki-9615 | Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Both Powers | :$\ds \int \frac {\sin^m a x} {\cos^n a x} \rd x = \frac {\sin^{m - 1} a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac {m - 1} {n - 1} \int \frac {\sin^{m - 2} a x} {\cos^{n - 2} a x} \rd x + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sin^{m - 1} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \paren {m - 1} a \sin^{m - 2} a x \cos a x
... | :$\ds \int \frac {\sin^m a x} {\cos^n a x} \rd x = \frac {\sin^{m - 1} a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac {m - 1} {n - 1} \int \frac {\sin^{m - 2} a x} {\cos^{n - 2} a x} \rd x + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sin^{m - 1} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \paren {m - 1} a ... | Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Both Powers | https://proofwiki.org/wiki/Primitive_of_Power_of_Sine_of_a_x_over_Power_of_Cosine_of_a_x/Reduction_of_Both_Powers | https://proofwiki.org/wiki/Primitive_of_Power_of_Sine_of_a_x_over_Power_of_Cosine_of_a_x/Reduction_of_Both_Powers | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Sine Function/Corollary",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Primitive of Power of Cosine of a x by Sine of a x",
"Integration by Parts"
] |
proofwiki-9616 | Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Power of Cosine | :$\ds \int \frac {\sin^m a x} {\cos^n a x} \rd x = \frac {\sin^{m + 1} a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac {m - n + 2} {n - 1} \int \frac {\sin^m a x} {\cos^{n - 2} a x} \rd x + C$ | {{begin-eqn}}
{{eqn | r = \frac {\sin^{m + 1} a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac {m + 1} {n - 1} \int \frac {\sin^m a x} {\cos^{n - 2} a x} \rd x + C
| o =
| c =
}}
{{eqn | r = \int \frac {\sin^{m + 2} a x} {\cos^n a x} \rd x
| c = Primitive of $\dfrac {\sin^m a x} {\cos^n a x}$: Reduc... | :$\ds \int \frac {\sin^m a x} {\cos^n a x} \rd x = \frac {\sin^{m + 1} a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac {m - n + 2} {n - 1} \int \frac {\sin^m a x} {\cos^{n - 2} a x} \rd x + C$ | {{begin-eqn}}
{{eqn | r = \frac {\sin^{m + 1} a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac {m + 1} {n - 1} \int \frac {\sin^m a x} {\cos^{n - 2} a x} \rd x + C
| o =
| c =
}}
{{eqn | r = \int \frac {\sin^{m + 2} a x} {\cos^n a x} \rd x
| c = [[Primitive of Power of Sine of a x over Power of Cosi... | Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Power of Cosine | https://proofwiki.org/wiki/Primitive_of_Power_of_Sine_of_a_x_over_Power_of_Cosine_of_a_x/Reduction_of_Power_of_Cosine | https://proofwiki.org/wiki/Primitive_of_Power_of_Sine_of_a_x_over_Power_of_Cosine_of_a_x/Reduction_of_Power_of_Cosine | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Both Powers",
"Sum of Squares of Sine and Cosine",
"Linear Combination of Integrals/Indefinite",
"Definition:Common Denominator"
] |
proofwiki-9617 | Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Power of Sine | :$\ds \int \frac {\sin^m a x} {\cos^n a x} \rd x = \frac {-\sin^{m - 1} a x} {a \paren {m - n} \cos^{n - 1} a x} + \frac {m - 1} {m - n} \int \frac {\sin^{m - 2} a x} {\cos^n a x} \rd x + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \frac {\sin^{m - 1} a x} {\cos^n a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {\cos^n a x \dfr... | :$\ds \int \frac {\sin^m a x} {\cos^n a x} \rd x = \frac {-\sin^{m - 1} a x} {a \paren {m - n} \cos^{n - 1} a x} + \frac {m - 1} {m - n} \int \frac {\sin^{m - 2} a x} {\cos^n a x} \rd x + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \frac {\sin^{m - 1} a x} {\cos^n a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| ... | Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Power of Sine | https://proofwiki.org/wiki/Primitive_of_Power_of_Sine_of_a_x_over_Power_of_Cosine_of_a_x/Reduction_of_Power_of_Sine | https://proofwiki.org/wiki/Primitive_of_Power_of_Sine_of_a_x_over_Power_of_Cosine_of_a_x/Reduction_of_Power_of_Sine | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Definition:Primitive",
"Quotient Rule for Derivatives",
"Derivative of Sine Function/Corollary",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Derivative of Cosine Function/Corollary",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Sum of Squares of Sine and... |
proofwiki-9618 | Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Both Powers | :$\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {-\cos^{m - 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m - 1} {n - 1} \int \frac {\cos^{m - 2} a x} {\sin^{n - 2} a x} \rd x + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \cos^{m - 1} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -\paren {m - 1} a \cos^{m - 2} a x \sin a x... | :$\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {-\cos^{m - 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m - 1} {n - 1} \int \frac {\cos^{m - 2} a x} {\sin^{n - 2} a x} \rd x + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \cos^{m - 1} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -\paren {m - 1} a... | Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Both Powers | https://proofwiki.org/wiki/Primitive_of_Power_of_Cosine_of_a_x_over_Power_of_Sine_of_a_x/Reduction_of_Both_Powers | https://proofwiki.org/wiki/Primitive_of_Power_of_Cosine_of_a_x_over_Power_of_Sine_of_a_x/Reduction_of_Both_Powers | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Cosine Function/Corollary",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Primitive of Power of Sine of a x by Cosine of a x",
"Integration by Parts"
] |
proofwiki-9619 | Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Power of Sine | :$\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m - n + 2} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C$ | {{begin-eqn}}
{{eqn | r = \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m + 1} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C
| o =
| c =
}}
{{eqn | r = \int \frac {\cos^{m + 2} a x} {\sin^n a x} \rd x
| c = Primitive of $\dfrac {\cos^m a x} {\sin^n a x}$: Redu... | :$\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m - n + 2} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C$ | {{begin-eqn}}
{{eqn | r = \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m + 1} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C
| o =
| c =
}}
{{eqn | r = \int \frac {\cos^{m + 2} a x} {\sin^n a x} \rd x
| c = [[Primitive of Power of Cosine of a x over Power of S... | Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Power of Sine | https://proofwiki.org/wiki/Primitive_of_Power_of_Cosine_of_a_x_over_Power_of_Sine_of_a_x/Reduction_of_Power_of_Sine | https://proofwiki.org/wiki/Primitive_of_Power_of_Cosine_of_a_x_over_Power_of_Sine_of_a_x/Reduction_of_Power_of_Sine | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Both Powers",
"Sum of Squares of Sine and Cosine",
"Linear Combination of Integrals/Indefinite",
"Definition:Fraction/Denominator"
] |
proofwiki-9620 | Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Power of Cosine | :$\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {\cos^{m - 1} a x} {a \paren {m - n} \sin^{n - 1} a x} + \frac {m - 1} {m - n} \int \frac {\cos^{m - 2} a x} {\sin^n a x} \rd x + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \frac {\cos^{m - 1} a x} {\sin^n a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {\sin^n a x \dfr... | :$\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {\cos^{m - 1} a x} {a \paren {m - n} \sin^{n - 1} a x} + \frac {m - 1} {m - n} \int \frac {\cos^{m - 2} a x} {\sin^n a x} \rd x + C$ | With a view to expressing the [[Definition:Primitive (Calculus)|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \frac {\cos^{m - 1} a x} {\sin^n a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d ... | Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Power of Cosine | https://proofwiki.org/wiki/Primitive_of_Power_of_Cosine_of_a_x_over_Power_of_Sine_of_a_x/Reduction_of_Power_of_Cosine | https://proofwiki.org/wiki/Primitive_of_Power_of_Cosine_of_a_x_over_Power_of_Sine_of_a_x/Reduction_of_Power_of_Cosine | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Definition:Primitive (Calculus)",
"Quotient Rule for Derivatives",
"Derivative of Cosine Function/Corollary",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Derivative of Sine Function/Corollary",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Sum of Squares ... |
proofwiki-9621 | Primitive of Reciprocal of Power of Cosine of a x by Power of Sine of a x/Reduction of Power of Cosine | :$\ds \int \frac {\d x} {\sin^m a x \cos^n a x} = \frac 1 {a \paren {n - 1} \sin^{m - 1} a x \cos^{n - 1} a x} + \frac {m + n - 2} {n - 1} \int \frac {\d x} {\sin^m a x \cos^{n - 2} a x}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin^m a x \cos^n a x}
| r = \int \frac {\sin^{-m} a x \rd x} {\cos^n a x}
| c =
}}
{{eqn | r = \frac {\sin^{-m + 1} a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac {-m - n + 2} {n - 1} \int \frac {\sin^{-m} a x} {\cos^{n - 2} a x} \rd x + C
| c = Primiti... | :$\ds \int \frac {\d x} {\sin^m a x \cos^n a x} = \frac 1 {a \paren {n - 1} \sin^{m - 1} a x \cos^{n - 1} a x} + \frac {m + n - 2} {n - 1} \int \frac {\d x} {\sin^m a x \cos^{n - 2} a x}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin^m a x \cos^n a x}
| r = \int \frac {\sin^{-m} a x \rd x} {\cos^n a x}
| c =
}}
{{eqn | r = \frac {\sin^{-m + 1} a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac {-m - n + 2} {n - 1} \int \frac {\sin^{-m} a x} {\cos^{n - 2} a x} \rd x + C
| c = [[Primi... | Primitive of Reciprocal of Power of Cosine of a x by Power of Sine of a x/Reduction of Power of Cosine | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Cosine_of_a_x_by_Power_of_Sine_of_a_x/Reduction_of_Power_of_Cosine | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Cosine_of_a_x_by_Power_of_Sine_of_a_x/Reduction_of_Power_of_Cosine | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Power of Cosine"
] |
proofwiki-9622 | Primitive of Reciprocal of Power of Cosine of a x by Power of Sine of a x/Reduction of Power of Sine | :$\ds \int \frac {\d x} {\sin^m a x \cos^n a x} = \frac {-1} {a \paren {n - 1} \sin^{m - 1} a x \cos^{n - 1} a x} + \frac {m + n - 2} {m - 1} \int \frac {\d x} {\sin^{m - 2} a x \cos^n a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin^m a x \cos^n a x}
| r = \int \frac {\cos^{-n} a x \rd x} {\sin^m a x}
| c =
}}
{{eqn | r = \frac {-\cos^{-n + 1} a x} {a \paren {m - 1} \sin^{m - 1} a x} - \frac {-n - m + 2} {m - 1} \int \frac {\cos^{-n} a x} {\sin^{m - 2} a x} \rd x + C
| c = Primit... | :$\ds \int \frac {\d x} {\sin^m a x \cos^n a x} = \frac {-1} {a \paren {n - 1} \sin^{m - 1} a x \cos^{n - 1} a x} + \frac {m + n - 2} {m - 1} \int \frac {\d x} {\sin^{m - 2} a x \cos^n a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin^m a x \cos^n a x}
| r = \int \frac {\cos^{-n} a x \rd x} {\sin^m a x}
| c =
}}
{{eqn | r = \frac {-\cos^{-n + 1} a x} {a \paren {m - 1} \sin^{m - 1} a x} - \frac {-n - m + 2} {m - 1} \int \frac {\cos^{-n} a x} {\sin^{m - 2} a x} \rd x + C
| c = [[Prim... | Primitive of Reciprocal of Power of Cosine of a x by Power of Sine of a x/Reduction of Power of Sine | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Cosine_of_a_x_by_Power_of_Sine_of_a_x/Reduction_of_Power_of_Sine | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Cosine_of_a_x_by_Power_of_Sine_of_a_x/Reduction_of_Power_of_Sine | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Power of Sine"
] |
proofwiki-9623 | Primitive of Tangent of a x/Cosine Form | :$\ds \int \tan a x \rd x = \frac {-\ln \size {\cos a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \tan x \rd x
| r = -\ln \size {\cos x}
| c = Primitive of $\tan x$: Cosine Form
}}
{{eqn | ll= \leadsto
| l = \int \tan a x \rd x
| r = \frac 1 a \paren {-\ln \size {\cos a x} } + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {-\ln \... | :$\ds \int \tan a x \rd x = \frac {-\ln \size {\cos a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \tan x \rd x
| r = -\ln \size {\cos x}
| c = [[Primitive of Tangent Function/Cosine Form|Primitive of $\tan x$: Cosine Form]]
}}
{{eqn | ll= \leadsto
| l = \int \tan a x \rd x
| r = \frac 1 a \paren {-\ln \size {\cos a x} } + C
| c = [[Primitive of Function o... | Primitive of Tangent of a x/Cosine Form | https://proofwiki.org/wiki/Primitive_of_Tangent_of_a_x/Cosine_Form | https://proofwiki.org/wiki/Primitive_of_Tangent_of_a_x/Cosine_Form | [
"Primitive of Tangent of a x"
] | [] | [
"Primitive of Tangent Function/Cosine Form",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9624 | Primitive of Tangent of a x/Secant Form | :$\ds \int \tan a x \rd x = \frac {\ln \size {\sec a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \tan x \rd x
| r = \ln \size {\sec x}
| c = Primitive of $\tan x$: Secant Form
}}
{{eqn | ll= \leadsto
| l = \int \tan a x \rd x
| r = \frac 1 a \paren {\ln \size {\sec a x} } + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\ln \siz... | :$\ds \int \tan a x \rd x = \frac {\ln \size {\sec a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \tan x \rd x
| r = \ln \size {\sec x}
| c = [[Primitive of Tangent Function/Secant Form|Primitive of $\tan x$: Secant Form]]
}}
{{eqn | ll= \leadsto
| l = \int \tan a x \rd x
| r = \frac 1 a \paren {\ln \size {\sec a x} } + C
| c = [[Primitive of Function of ... | Primitive of Tangent of a x/Secant Form | https://proofwiki.org/wiki/Primitive_of_Tangent_of_a_x/Secant_Form | https://proofwiki.org/wiki/Primitive_of_Tangent_of_a_x/Secant_Form | [
"Primitive of Tangent of a x"
] | [] | [
"Primitive of Tangent Function/Secant Form",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9625 | Primitive of Square of Tangent of a x | :$\ds \int \tan^2 a x \rd x = \frac {\tan a x} a - x + C$ | {{begin-eqn}}
{{eqn | l = \int \tan^2 x \rd x
| r = \tan x - x + C
| c = Primitive of $\tan^2 x$
}}
{{eqn | ll= \leadsto
| l = \int \tan^2 a x \rd x
| r = \frac 1 a \paren {\tan a x - a x} + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\tan a x} a - x + C
... | :$\ds \int \tan^2 a x \rd x = \frac {\tan a x} a - x + C$ | {{begin-eqn}}
{{eqn | l = \int \tan^2 x \rd x
| r = \tan x - x + C
| c = [[Primitive of Square of Tangent Function|Primitive of $\tan^2 x$]]
}}
{{eqn | ll= \leadsto
| l = \int \tan^2 a x \rd x
| r = \frac 1 a \paren {\tan a x - a x} + C
| c = [[Primitive of Function of Constant Multiple]]
... | Primitive of Square of Tangent of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Tangent_of_a_x | [
"Primitives involving Tangent Function"
] | [] | [
"Primitive of Square of Tangent Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9626 | Primitive of Cube of Tangent of a x | :$\ds \int \tan^3 a x \rd x = \frac {\tan^2 a x} {2 a} + \frac 1 a \ln \size {\cos a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \tan^3 a x \rd x
| r = \int \tan a x \tan^2 a x \rd x
| c =
}}
{{eqn | r = \int \tan a x \paren {\sec^2 a x - 1} \rd x
| c = Difference of Squares of Secant and Tangent
}}
{{eqn | r = \int \tan a x \sec^2 a x \rd x - \int \tan a x \rd x
| c = Linear Combination of... | :$\ds \int \tan^3 a x \rd x = \frac {\tan^2 a x} {2 a} + \frac 1 a \ln \size {\cos a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \tan^3 a x \rd x
| r = \int \tan a x \tan^2 a x \rd x
| c =
}}
{{eqn | r = \int \tan a x \paren {\sec^2 a x - 1} \rd x
| c = [[Difference of Squares of Secant and Tangent]]
}}
{{eqn | r = \int \tan a x \sec^2 a x \rd x - \int \tan a x \rd x
| c = [[Linear Combinat... | Primitive of Cube of Tangent of a x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Cube_of_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Tangent_of_a_x/Proof_1 | [
"Primitive of Cube of Tangent of a x",
"Primitives involving Tangent Function"
] | [] | [
"Sum of Squares of Sine and Cosine/Corollary 1",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power of Tangent of a x by Square of Secant of a x",
"Primitive of Tangent of a x/Cosine Form"
] |
proofwiki-9627 | Primitive of Cube of Tangent of a x | :$\ds \int \tan^3 a x \rd x = \frac {\tan^2 a x} {2 a} + \frac 1 a \ln \size {\cos a x} + C$ | {{begin-eqn}}
{{eqn | l = I_n
| r = \int \map {\tan^n} {a x} \rd x
}}
{{eqn | r = \frac {\map {\tan^{n - 1} } {a x} } {a \paren {n - 1} } - I_{n - 2}
| c = Reduction Formula for Integral of Power of Tangent
}}
{{eqn | l = I_1
| r = -\frac 1 a \ln \size {\map \cos {a x} } + C
| c = Primitive of ... | :$\ds \int \tan^3 a x \rd x = \frac {\tan^2 a x} {2 a} + \frac 1 a \ln \size {\cos a x} + C$ | {{begin-eqn}}
{{eqn | l = I_n
| r = \int \map {\tan^n} {a x} \rd x
}}
{{eqn | r = \frac {\map {\tan^{n - 1} } {a x} } {a \paren {n - 1} } - I_{n - 2}
| c = [[Reduction Formula for Integral of Power of Tangent]]
}}
{{eqn | l = I_1
| r = -\frac 1 a \ln \size {\map \cos {a x} } + C
| c = [[Primiti... | Primitive of Cube of Tangent of a x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Cube_of_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Tangent_of_a_x/Proof_2 | [
"Primitive of Cube of Tangent of a x",
"Primitives involving Tangent Function"
] | [] | [
"Reduction Formula for Integral of Power of Tangent",
"Primitive of Tangent of a x/Cosine Form"
] |
proofwiki-9628 | Primitive of Power of Tangent of a x by Square of Secant of a x | :$\ds \int \tan^n a x \sec^2 a x \rd x = \frac {\tan^{n + 1} a x} {\paren {n + 1} a} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \tan a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a \sec^2 a x
| c = Derivative of $\tan a x$
}}
{{eqn | ll= \leadsto
| l = \int \tan^n a x \sec^2 a x \rd x
| r = \int \frac 1 a z^n \rd z
| c = Integration by Substit... | :$\ds \int \tan^n a x \sec^2 a x \rd x = \frac {\tan^{n + 1} a x} {\paren {n + 1} a} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \tan a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a \sec^2 a x
| c = [[Derivative of Tangent of a x|Derivative of $\tan a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \tan^n a x \sec^2 a x \rd x
| r = \int \frac 1 a z^n \rd z
... | Primitive of Power of Tangent of a x by Square of Secant of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Tangent_of_a_x_by_Square_of_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Tangent_of_a_x_by_Square_of_Secant_of_a_x | [
"Primitives involving Tangent Function",
"Primitives involving Secant Function"
] | [] | [
"Derivative of Tangent Function/Corollary 1",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9629 | Primitive of Cotangent of a x | :$\ds \int \cot a x \rd x = \frac {\ln \size {\sin a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \cot x \rd x
| r = \ln \size {\sin x}
| c = Primitive of $\cot x$
}}
{{eqn | ll= \leadsto
| l = \int \cot a x \rd x
| r = \frac 1 a \paren {\ln \size {\sin a x} } + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\ln \size {\sin a x} ... | :$\ds \int \cot a x \rd x = \frac {\ln \size {\sin a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \cot x \rd x
| r = \ln \size {\sin x}
| c = [[Primitive of Cotangent Function|Primitive of $\cot x$]]
}}
{{eqn | ll= \leadsto
| l = \int \cot a x \rd x
| r = \frac 1 a \paren {\ln \size {\sin a x} } + C
| c = [[Primitive of Function of Constant Multiple]]
}}
... | Primitive of Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cotangent_of_a_x | [
"Primitive of Cotangent of a x",
"Primitive of Cotangent Function",
"Primitives involving Cotangent Function"
] | [] | [
"Primitive of Cotangent Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9630 | Primitive of Reciprocal of Tangent of a x | :$\ds \int \frac {\d x} {\tan a x} = \frac 1 a \ln \size {\sin a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\tan a x}
| r = \int \cot a x \rd x
| c = Cotangent is Reciprocal of Tangent
}}
{{eqn | r = \frac 1 a \ln \size {\sin a x} + C
| c = Primitive of $\cot a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {\tan a x} = \frac 1 a \ln \size {\sin a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\tan a x}
| r = \int \cot a x \rd x
| c = [[Cotangent is Reciprocal of Tangent]]
}}
{{eqn | r = \frac 1 a \ln \size {\sin a x} + C
| c = [[Primitive of Cotangent of a x|Primitive of $\cot a x$]]
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of Tangent of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Tangent_of_a_x | [
"Primitives involving Tangent Function"
] | [] | [
"Cotangent is Reciprocal of Tangent",
"Primitive of Cotangent of a x"
] |
proofwiki-9631 | Primitive of x by Tangent of a x | :$\ds \int x \tan a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x} ^ 3} 3 + \frac {\paren {a x}^5} {15} + \frac {2 \paren {a x}^7} {105} + \cdots + \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
where $B_{2 n}$ denotes the $2 n$th Bernoul... | From Power Series Expansion for Tangent Function:
{{:Power Series Expansion for Tangent Function}}
{{begin-eqn}}
{{eqn | l = x \tan ax
| r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} a^{2 n - 1} x^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | ll= \leadsto
|... | :$\ds \int x \tan a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x} ^ 3} 3 + \frac {\paren {a x}^5} {15} + \frac {2 \paren {a x}^7} {105} + \cdots + \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
where $B_{2 n}$ denotes the $2 n$th [[Defi... | From [[Power Series Expansion for Tangent Function]]:
{{:Power Series Expansion for Tangent Function}}
{{begin-eqn}}
{{eqn | l = x \tan ax
| r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} a^{2 n - 1} x^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | ll= \leadsto
... | Primitive of x by Tangent of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Tangent_of_a_x | [
"Primitives involving Tangent Function"
] | [
"Definition:Bernoulli Numbers"
] | [
"Power Series Expansion for Tangent Function",
"Primitive of Power"
] |
proofwiki-9632 | Primitive of Tangent of a x over x | :$\ds \int \frac {\tan a x} x \rd x = a x + \frac {\paren {a x}^3} 9 + \frac {2 \paren {a x}^5} {75} + \cdots + \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + \cdots + C$
where $B_n$ denotes the $n$th Bernoulli number. | {{begin-eqn}}
{{eqn | l = \int \frac {\tan a x} x \rd x
| r = \int \frac 1 x \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n}!} \rd x
| c = Power Series Expansion for Tangent Function
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty... | :$\ds \int \frac {\tan a x} x \rd x = a x + \frac {\paren {a x}^3} 9 + \frac {2 \paren {a x}^5} {75} + \cdots + \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + \cdots + C$
where $B_n$ denotes the $n$th [[Definition:Bernoulli Numbers|Bernoulli ... | {{begin-eqn}}
{{eqn | l = \int \frac {\tan a x} x \rd x
| r = \int \frac 1 x \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n}!} \rd x
| c = [[Power Series Expansion for Tangent Function]]
}}
{{eqn | r = \sum_{n \mathop = 1}^\i... | Primitive of Tangent of a x over x | https://proofwiki.org/wiki/Primitive_of_Tangent_of_a_x_over_x | https://proofwiki.org/wiki/Primitive_of_Tangent_of_a_x_over_x | [
"Primitives involving Tangent Function"
] | [
"Definition:Bernoulli Numbers"
] | [
"Power Series Expansion for Tangent Function",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9633 | Primitive of x by Square of Tangent of a x | :$\ds \int x \tan^2 a x \rd x = \frac {x \tan a x} a + \frac 1 {a^2} \ln \size {\cos a x} - \frac {x^2} 2 + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Primitive of Power
}}
{{end-eqn}}
and let:
{... | :$\ds \int x \tan^2 a x \rd x = \frac {x \tan a x} a + \frac 1 {a^2} \ln \size {\cos a x} - \frac {x^2} 2 + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Primitive of Pow... | Primitive of x by Square of Tangent of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Tangent_of_a_x | [
"Primitives involving Tangent Function"
] | [] | [
"Definition:Primitive",
"Primitive of Power",
"Primitive of Square of Tangent of a x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Tangent of a x/Cosine Form",
"Primitive of Power"
] |
proofwiki-9634 | Primitive of Reciprocal of p plus q by Tangent of a x | :$\ds \int \frac {\d x} {p + q \tan a x} = \frac {p x} {p^2 + q^2} + \frac q {a \paren {p^2 + q^2} } \ln \size {q \sin a x + p \cos a x} + C$ | First, let $\arctan \dfrac p q = \phi$.
Let $z = a x + \phi$.
{{begin-eqn}}
{{eqn | l = z
| r = \map \sin {a x + \phi}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a \map \cos {a x + \phi}
| c = Derivative of $\sin a x$ etc.
}}
{{eqn | r = a \cos z
| c =
}}
{{end-e... | :$\ds \int \frac {\d x} {p + q \tan a x} = \frac {p x} {p^2 + q^2} + \frac q {a \paren {p^2 + q^2} } \ln \size {q \sin a x + p \cos a x} + C$ | First, let $\arctan \dfrac p q = \phi$.
Let $z = a x + \phi$.
{{begin-eqn}}
{{eqn | l = z
| r = \map \sin {a x + \phi}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a \map \cos {a x + \phi}
| c = [[Derivative of Sine of a x|Derivative of $\sin a x$]] etc.
}}
{{eqn | r = ... | Primitive of Reciprocal of p plus q by Tangent of a x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Tangent_of_a_x/Proof_1 | [
"Primitives involving Tangent Function",
"Primitive of Reciprocal of p plus q by Tangent of a x"
] | [] | [
"Derivative of Sine Function/Corollary",
"Tangent is Sine divided by Cosine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Multiple of Sine plus Multiple of Cosine/Sine Form",
"Primitive of Constant Multiple of Function",
"Primitive of Cosine of a x over Sine of a x plus phi"
] |
proofwiki-9635 | Primitive of Reciprocal of p plus q by Tangent of a x | :$\ds \int \frac {\d x} {p + q \tan a x} = \frac {p x} {p^2 + q^2} + \frac q {a \paren {p^2 + q^2} } \ln \size {q \sin a x + p \cos a x} + C$ | We have:
:$\dfrac \d {\d x} \paren {q \sin a x + p \cos a x} = a q \cos a x - a p \sin a x$
Thus:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \tan a x}
| r = \int \frac {\d x} {p + q \dfrac {\sin a x} {\cos a x} }
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \int \frac {\cos a x \rd x} {p \... | :$\ds \int \frac {\d x} {p + q \tan a x} = \frac {p x} {p^2 + q^2} + \frac q {a \paren {p^2 + q^2} } \ln \size {q \sin a x + p \cos a x} + C$ | We have:
:$\dfrac \d {\d x} \paren {q \sin a x + p \cos a x} = a q \cos a x - a p \sin a x$
Thus:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \tan a x}
| r = \int \frac {\d x} {p + q \dfrac {\sin a x} {\cos a x} }
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \int \frac {\cos a x \rd x... | Primitive of Reciprocal of p plus q by Tangent of a x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Tangent_of_a_x/Proof_2 | [
"Primitives involving Tangent Function",
"Primitive of Reciprocal of p plus q by Tangent of a x"
] | [] | [
"Tangent is Sine divided by Cosine",
"Primitive of Constant",
"Primitive of Reciprocal"
] |
proofwiki-9636 | Primitive of Power of Tangent of a x | :$\ds \int \tan^n a x \rd x = \frac {\tan^{n - 1} a x} {\paren {n - 1} a} - \int \tan^{n - 2} a x \rd x$
for $n \ne 1$. | {{begin-eqn}}
{{eqn | l = \int \tan^n a x \rd x
| r = \int \tan^{n - 2} a x \tan^2 a x \rd x
| c =
}}
{{eqn | r = \int \tan^{n - 2} a x \paren {\sec^2 a x - 1} \rd x
| c = Difference of Squares of Secant and Tangent
}}
{{eqn | r = \int \tan^{n - 2} a x \sec^2 a x \rd x - \int \tan^{n - 2} \rd x
... | :$\ds \int \tan^n a x \rd x = \frac {\tan^{n - 1} a x} {\paren {n - 1} a} - \int \tan^{n - 2} a x \rd x$
for $n \ne 1$. | {{begin-eqn}}
{{eqn | l = \int \tan^n a x \rd x
| r = \int \tan^{n - 2} a x \tan^2 a x \rd x
| c =
}}
{{eqn | r = \int \tan^{n - 2} a x \paren {\sec^2 a x - 1} \rd x
| c = [[Difference of Squares of Secant and Tangent]]
}}
{{eqn | r = \int \tan^{n - 2} a x \sec^2 a x \rd x - \int \tan^{n - 2} \rd x
... | Primitive of Power of Tangent of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Tangent_of_a_x | [
"Primitives involving Tangent Function"
] | [] | [
"Sum of Squares of Sine and Cosine/Corollary 1",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power of Tangent of a x by Square of Secant of a x"
] |
proofwiki-9637 | Primitive of Square of Cotangent of a x | :$\ds \int \cot^2 a x \rd x = \frac {-\cot a x} a - x + C$ | {{begin-eqn}}
{{eqn | l = \int \cot^2 x \rd x
| r = -\cot x - x
| c = Primitive of $\cot^2 x$
}}
{{eqn | ll= \leadsto
| l = \int \cot a x \rd x
| r = \frac 1 a \paren {-\cot a x - a x} + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {-\cot a x} a - x + C
| c... | :$\ds \int \cot^2 a x \rd x = \frac {-\cot a x} a - x + C$ | {{begin-eqn}}
{{eqn | l = \int \cot^2 x \rd x
| r = -\cot x - x
| c = [[Primitive of Square of Cotangent Function|Primitive of $\cot^2 x$]]
}}
{{eqn | ll= \leadsto
| l = \int \cot a x \rd x
| r = \frac 1 a \paren {-\cot a x - a x} + C
| c = [[Primitive of Function of Constant Multiple]]
}}... | Primitive of Square of Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Cotangent_of_a_x | [
"Primitives involving Cotangent Function"
] | [] | [
"Primitive of Square of Cotangent Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9638 | Primitive of Cube of Cotangent of a x | :$\ds \int \cot^3 a x \rd x = \frac {-\cot^2 a x} {2 a} - \frac 1 a \ln \size {\sin a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \cot^3 x \rd x
| r = \int \cot a x \cot^2 a x \rd x
| c =
}}
{{eqn | r = \int \cot a x \paren {\csc^2 a x - 1} \rd x
| c = Difference of Squares of Cosecant and Cotangent
}}
{{eqn | r = \int \cot a x \csc^2 a x \rd x - \int \cot a x \rd x
| c = Linear Combination ... | :$\ds \int \cot^3 a x \rd x = \frac {-\cot^2 a x} {2 a} - \frac 1 a \ln \size {\sin a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \cot^3 x \rd x
| r = \int \cot a x \cot^2 a x \rd x
| c =
}}
{{eqn | r = \int \cot a x \paren {\csc^2 a x - 1} \rd x
| c = [[Difference of Squares of Cosecant and Cotangent]]
}}
{{eqn | r = \int \cot a x \csc^2 a x \rd x - \int \cot a x \rd x
| c = [[Linear Combin... | Primitive of Cube of Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Cotangent_of_a_x | [
"Primitives involving Cotangent Function"
] | [] | [
"Sum of Squares of Sine and Cosine/Corollary 2",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power of Cotangent of a x by Square of Cosecant of a x",
"Primitive of Cotangent of a x"
] |
proofwiki-9639 | Primitive of Power of Cotangent of a x by Square of Cosecant of a x | :$\ds \int \cot^n a x \csc^2 a x \rd x = \frac {-\cot^{n + 1} a x} {\paren {n + 1} a} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \cot a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -a \csc^2 a x
| c = Derivative of $\cot a x$
}}
{{eqn | ll= \leadsto
| l = \int \cot^n a x \csc^2 a x \rd x
| r = \int \frac {-1} a z^n \rd z
| c = Integration by Sub... | :$\ds \int \cot^n a x \csc^2 a x \rd x = \frac {-\cot^{n + 1} a x} {\paren {n + 1} a} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \cot a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -a \csc^2 a x
| c = [[Derivative of Cotangent of a x|Derivative of $\cot a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \cot^n a x \csc^2 a x \rd x
| r = \int \frac {-1} a z^n \... | Primitive of Power of Cotangent of a x by Square of Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Cotangent_of_a_x_by_Square_of_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Cotangent_of_a_x_by_Square_of_Cosecant_of_a_x | [
"Primitives involving Cotangent Function",
"Primitives involving Cosecant Function"
] | [] | [
"Derivative of Cotangent Function/Corollary 1",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9640 | Primitive of Square of Cosecant of a x over Cotangent of a x | :$\ds \int \frac {\csc^2 a x \rd x} {\cot a x} = \frac {-\ln \size {\cot a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \frac \d {\d x} \cot x
| r = -\csc^2 x
| c = Derivative of Cotangent Function
}}
{{eqn | ll= \leadsto
| l = \int \frac {\csc^2 x \rd x} {\cot x}
| r = -\ln \size {\cot a x} + C
| c = Primitive of Function under its Derivative
}}
{{eqn | ll= \leadsto
| l = \i... | :$\ds \int \frac {\csc^2 a x \rd x} {\cot a x} = \frac {-\ln \size {\cot a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \frac \d {\d x} \cot x
| r = -\csc^2 x
| c = [[Derivative of Cotangent Function]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\csc^2 x \rd x} {\cot x}
| r = -\ln \size {\cot a x} + C
| c = [[Primitive of Function under its Derivative]]
}}
{{eqn | ll= \leadsto
... | Primitive of Square of Cosecant of a x over Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosecant_of_a_x_over_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosecant_of_a_x_over_Cotangent_of_a_x | [
"Primitives involving Cotangent Function",
"Primitives involving Cosecant Function"
] | [] | [
"Derivative of Cotangent Function",
"Primitive of Function under its Derivative",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9641 | Primitive of Reciprocal of Cotangent of a x | :$\ds \int \frac {\d x} {\cot a x} = \frac {-\ln \size {\cos a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cot a x}
| r = \int \tan a x \rd x
| c = Cotangent is Reciprocal of Tangent
}}
{{eqn | r = \frac {-\ln \size {\cos a x} } a + C
| c = Primitive of $\tan a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {\cot a x} = \frac {-\ln \size {\cos a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cot a x}
| r = \int \tan a x \rd x
| c = [[Cotangent is Reciprocal of Tangent]]
}}
{{eqn | r = \frac {-\ln \size {\cos a x} } a + C
| c = [[Primitive of Tangent of a x/Cosine Form|Primitive of $\tan a x$]]
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cotangent_of_a_x | [
"Primitives involving Cotangent Function"
] | [] | [
"Cotangent is Reciprocal of Tangent",
"Primitive of Tangent of a x/Cosine Form"
] |
proofwiki-9642 | Primitive of x by Cotangent of a x | :$\ds \int x \cot a x \rd x = \frac 1 {a ^ 2} \paren {a x - \frac {\paren {a x}^3} 9 - \frac {\paren {a x}^5} {225} - \cdots + \frac {\paren {-1}^n 2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1} !} + \cdots} + C$
where $B_{2 n}$ denotes the $2 n$th Bernoulli number. | From Power Series Expansion for Cotangent Function:
{{:Power Series Expansion for Cotangent Function}}
{{begin-eqn}}
{{eqn | l = x \cot ax
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} a^{2 n - 1} x^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | ll= \leadsto
| l = \int x \cot a x \r... | :$\ds \int x \cot a x \rd x = \frac 1 {a ^ 2} \paren {a x - \frac {\paren {a x}^3} 9 - \frac {\paren {a x}^5} {225} - \cdots + \frac {\paren {-1}^n 2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1} !} + \cdots} + C$
where $B_{2 n}$ denotes the $2 n$th [[Definition:Bernoulli Numbers|Bernoulli number]]. | From [[Power Series Expansion for Cotangent Function]]:
{{:Power Series Expansion for Cotangent Function}}
{{begin-eqn}}
{{eqn | l = x \cot ax
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} a^{2 n - 1} x^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | ll= \leadsto
| l = \int x \cot ... | Primitive of x by Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Cotangent_of_a_x | [
"Primitives involving Cotangent Function"
] | [
"Definition:Bernoulli Numbers"
] | [
"Power Series Expansion for Cotangent Function",
"Primitive of Power"
] |
proofwiki-9643 | Primitive of Cotangent of a x over x | :$\ds \int \frac {\cot a x} x \rd x = \frac {-1} a x - \frac {a x} 3 - \frac {\paren {a x}^3} {135} - \cdots - \frac {\paren {-1}^{n - 1} 2^{2 n} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} - \cdots + C$
where $B_n$ denotes the $n$th Bernoulli number. | {{begin-eqn}}
{{eqn | l = \int \frac {\csc a x} x \rd x
| r = \int \frac 1 x \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n}!} \rd x
| c = Power Series Expansion for Cosecant Function
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2... | :$\ds \int \frac {\cot a x} x \rd x = \frac {-1} a x - \frac {a x} 3 - \frac {\paren {a x}^3} {135} - \cdots - \frac {\paren {-1}^{n - 1} 2^{2 n} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} - \cdots + C$
where $B_n$ denotes the $n$th [[Definition:Bernoulli Numbers|Bernoulli number]]. | {{begin-eqn}}
{{eqn | l = \int \frac {\csc a x} x \rd x
| r = \int \frac 1 x \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n}!} \rd x
| c = [[Power Series Expansion for Cosecant Function]]
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n ... | Primitive of Cotangent of a x over x | https://proofwiki.org/wiki/Primitive_of_Cotangent_of_a_x_over_x | https://proofwiki.org/wiki/Primitive_of_Cotangent_of_a_x_over_x | [
"Primitives involving Cotangent Function"
] | [
"Definition:Bernoulli Numbers"
] | [
"Power Series Expansion for Cosecant Function",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9644 | Primitive of x by Square of Cotangent of a x | :$\ds \int x \cot^2 a x \rd x = \frac {-x \cot a x} a + \frac 1 {a^2} \ln \size {\sin a x} - \frac {x^2} 2 + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Primitive of Power
}}
{{end-eqn}}
and let:
{... | :$\ds \int x \cot^2 a x \rd x = \frac {-x \cot a x} a + \frac 1 {a^2} \ln \size {\sin a x} - \frac {x^2} 2 + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Primitive of Pow... | Primitive of x by Square of Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Cotangent_of_a_x | [
"Primitives involving Cotangent Function"
] | [] | [
"Definition:Primitive",
"Primitive of Power",
"Primitive of Square of Cotangent of a x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Cotangent of a x",
"Primitive of Power"
] |
proofwiki-9645 | Primitive of Reciprocal of p plus q by Cotangent of a x | :$\ds \int \frac {\d x} {p + q \cot a x} = \frac {p x} {p^2 + q^2} - \frac q {a \paren {p^2 + q^2} } \ln \size {p \sin a x + q \cos a x} + C$ | We have:
:$\dfrac \d {\d x} \paren {p \sin a x + q \cos a x} = a p \cos a x - a q \sin a x$
Thus:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \cot a x}
| r = \int \frac {\d x} {p + q \dfrac {\cos a x} {\sin a x} }
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \int \frac {\sin a x \rd x} {p... | :$\ds \int \frac {\d x} {p + q \cot a x} = \frac {p x} {p^2 + q^2} - \frac q {a \paren {p^2 + q^2} } \ln \size {p \sin a x + q \cos a x} + C$ | We have:
:$\dfrac \d {\d x} \paren {p \sin a x + q \cos a x} = a p \cos a x - a q \sin a x$
Thus:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \cot a x}
| r = \int \frac {\d x} {p + q \dfrac {\cos a x} {\sin a x} }
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \int \frac {\sin a x \rd... | Primitive of Reciprocal of p plus q by Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Cotangent_of_a_x | [
"Primitives involving Cotangent Function"
] | [] | [
"Cotangent is Cosine divided by Sine",
"Primitive of Constant",
"Primitive of Reciprocal"
] |
proofwiki-9646 | Primitive of Power of Cotangent of a x | :$\ds \int \cot^n a x \rd x = \frac {-\cot^{n - 1} a x} {\paren {n - 1} a} - \int \cot^{n - 2} a x \rd x$
for $n \ne 1$. | {{begin-eqn}}
{{eqn | l = \int \cot^n a x \rd x
| r = \int \cot^{n - 2} a x \cot^2 a x \rd x
| c =
}}
{{eqn | r = \int \cot^{n - 2} a x \paren {\csc^2 a x - 1} \rd x
| c = Difference of Squares of Cosecant and Cotangent
}}
{{eqn | r = \int \cot^{n - 2} a x \csc^2 a x \rd x - \int \cot^{n - 2} \rd x
... | :$\ds \int \cot^n a x \rd x = \frac {-\cot^{n - 1} a x} {\paren {n - 1} a} - \int \cot^{n - 2} a x \rd x$
for $n \ne 1$. | {{begin-eqn}}
{{eqn | l = \int \cot^n a x \rd x
| r = \int \cot^{n - 2} a x \cot^2 a x \rd x
| c =
}}
{{eqn | r = \int \cot^{n - 2} a x \paren {\csc^2 a x - 1} \rd x
| c = [[Difference of Squares of Cosecant and Cotangent]]
}}
{{eqn | r = \int \cot^{n - 2} a x \csc^2 a x \rd x - \int \cot^{n - 2} \rd... | Primitive of Power of Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Cotangent_of_a_x | [
"Primitives involving Cotangent Function"
] | [] | [
"Sum of Squares of Sine and Cosine/Corollary 2",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power of Cotangent of a x by Square of Cosecant of a x"
] |
proofwiki-9647 | Primitive of Cube of Secant of a x | :$\ds \int \sec^3 a x \rd x = \frac 1 {2 a} \paren {\sec a x \tan a x + \ln \size {\sec a x + \tan a x} } + C$ | {{begin-eqn}}
{{eqn | l = \int \sec^3 x \rd x
| r = \frac {\sec a x \tan a x} {2 a} + \frac 1 2 \int \sec a x \rd x
| c = Primitive of $\sec^n a x$ where $n = 3$
}}
{{eqn | r = \frac {\sec a x \tan a x} {2 a} + \frac 1 2 \paren {\frac 1 a \ln \size {\sec a x + \tan a x} }
| c = Primitive of $\sec a x$... | :$\ds \int \sec^3 a x \rd x = \frac 1 {2 a} \paren {\sec a x \tan a x + \ln \size {\sec a x + \tan a x} } + C$ | {{begin-eqn}}
{{eqn | l = \int \sec^3 x \rd x
| r = \frac {\sec a x \tan a x} {2 a} + \frac 1 2 \int \sec a x \rd x
| c = [[Primitive of Power of Secant of a x|Primitive of $\sec^n a x$]] where $n = 3$
}}
{{eqn | r = \frac {\sec a x \tan a x} {2 a} + \frac 1 2 \paren {\frac 1 a \ln \size {\sec a x + \tan a ... | Primitive of Cube of Secant of a x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Cube_of_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Secant_of_a_x/Proof_1 | [
"Primitive of Cube of Secant Function",
"Primitives involving Secant Function"
] | [] | [
"Primitive of Power of Secant of a x",
"Primitive of Secant of a x/Secant plus Tangent Form"
] |
proofwiki-9648 | Primitive of Cube of Secant of a x | :$\ds \int \sec^3 a x \rd x = \frac 1 {2 a} \paren {\sec a x \tan a x + \ln \size {\sec a x + \tan a x} } + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sec a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \sec a x \tan a x
| c = Derivative of Funct... | :$\ds \int \sec^3 a x \rd x = \frac 1 {2 a} \paren {\sec a x \tan a x + \ln \size {\sec a x + \tan a x} } + C$ | With a view to expressing the [[Definition:Primitive (Calculus)|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sec a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \sec a x \ta... | Primitive of Cube of Secant of a x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Cube_of_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Secant_of_a_x/Proof_2 | [
"Primitive of Cube of Secant Function",
"Primitives involving Secant Function"
] | [] | [
"Definition:Primitive (Calculus)",
"Derivative of Function of Constant Multiple",
"Derivative of Secant Function",
"Primitive of Square of Secant of a x",
"Integration by Parts",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Linear Combination of Integrals/Indefinite",
"Primitive of Secant of a x/... |
proofwiki-9649 | Primitive of Reciprocal of Secant of a x | :$\ds \int \frac {\d x} {\sec a x} = \frac {\sin a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sec a x}
| r = \int \cos a x \rd x
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac {\sin a x} a + C
| c = Primitive of $\cos a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {\sec a x} = \frac {\sin a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sec a x}
| r = \int \cos a x \rd x
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac {\sin a x} a + C
| c = [[Primitive of Cosine of a x|Primitive of $\cos a x$]]
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of Secant of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Secant_of_a_x | [
"Primitives involving Secant Function"
] | [] | [
"Secant is Reciprocal of Cosine",
"Primitive of Cosine Function/Corollary"
] |
proofwiki-9650 | Primitive of x by Secant of a x | {{begin-eqn}}
{{eqn | l = \int x \sec a x \rd x
| r = \frac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 - \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} - \cdots + \frac {\paren {-1}^n E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + \cdots} + C
| c =
}}
{{eqn | r = \frac 1 {a^2}... | {{begin-eqn}}
{{eqn | l = \int x \sec a x \rd x
| r = \frac 1 {a^2} \int \theta \sec \theta \rd \theta
| c = Substitution of $a x \to \theta$
}}
{{eqn | r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{ \paren {-1}^n E_{2 n} \theta^{2 n} } {\paren {2 n}!} \rd \theta
| c = Power Series Expansion fo... | {{begin-eqn}}
{{eqn | l = \int x \sec a x \rd x
| r = \frac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 - \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} - \cdots + \frac {\paren {-1}^n E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + \cdots} + C
| c =
}}
{{eqn | r = \frac 1 {a^2}... | {{begin-eqn}}
{{eqn | l = \int x \sec a x \rd x
| r = \frac 1 {a^2} \int \theta \sec \theta \rd \theta
| c = [[Integration by Substitution|Substitution of $a x \to \theta$]]
}}
{{eqn | r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{ \paren {-1}^n E_{2 n} \theta^{2 n} } {\paren {2 n}!} \rd \theta
... | Primitive of x by Secant of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Secant_of_a_x | [
"Primitives involving Secant Function"
] | [
"Definition:Euler Numbers"
] | [
"Integration by Substitution",
"Power Series Expansion for Secant Function",
"Power Series is Termwise Integrable within Radius of Convergence",
"Integration by Substitution"
] |
proofwiki-9651 | Primitive of Secant of a x over x | :$\ds \int \frac {\sec a x} x \rd x = \ln \size x + \frac {\paren {a x}^2} 4 + \frac {5 \paren {a x}^4} {96} + \frac {61 \paren {a x}^6} {4320} + \cdots + \frac {\paren {-1}^n E_n \paren {a x}^{2 n} } {\paren {2 n} \paren {2 n}!} + \cdots + C$
where $E_n$ is the $n$th Euler number. | {{begin-eqn}}
{{eqn | l = \int \frac {\sec a x} x \rd x
| r = \int \frac 1 x \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n E_{2 n} \paren {a x}^{2 n} } {\paren {2 n}!} \rd x
| c = Power Series Expansion for Secant Function
}}
{{eqn | r = \int \frac {E_0} x \rd x + \sum_{n \mathop = 1}^\infty \frac {\pare... | :$\ds \int \frac {\sec a x} x \rd x = \ln \size x + \frac {\paren {a x}^2} 4 + \frac {5 \paren {a x}^4} {96} + \frac {61 \paren {a x}^6} {4320} + \cdots + \frac {\paren {-1}^n E_n \paren {a x}^{2 n} } {\paren {2 n} \paren {2 n}!} + \cdots + C$
where $E_n$ is the $n$th [[Definition:Euler Numbers|Euler number]]. | {{begin-eqn}}
{{eqn | l = \int \frac {\sec a x} x \rd x
| r = \int \frac 1 x \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n E_{2 n} \paren {a x}^{2 n} } {\paren {2 n}!} \rd x
| c = [[Power Series Expansion for Secant Function]]
}}
{{eqn | r = \int \frac {E_0} x \rd x + \sum_{n \mathop = 1}^\infty \frac {\... | Primitive of Secant of a x over x | https://proofwiki.org/wiki/Primitive_of_Secant_of_a_x_over_x | https://proofwiki.org/wiki/Primitive_of_Secant_of_a_x_over_x | [
"Primitives involving Secant Function"
] | [
"Definition:Euler Numbers"
] | [
"Power Series Expansion for Secant Function",
"Primitive of Constant Multiple of Function",
"Primitive of Power",
"Primitive of Reciprocal"
] |
proofwiki-9652 | Primitive of x by Square of Secant of a x | :$\ds \int x \sec^2 a x \rd x = \frac {x \tan a x} a + \frac 1 {a^2} \ln \size {\cos a x} + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int x \sec^2 a x \rd x = \frac {x \tan a x} a + \frac 1 {a^2} \ln \size {\cos a x} + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x by Square of Secant of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Secant_of_a_x | [
"Primitives involving Secant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Square of Secant of a x",
"Integration by Parts",
"Primitive of Tangent of a x/Cosine Form"
] |
proofwiki-9653 | Primitive of Reciprocal of q plus p by Secant of a x | :$\ds \int \frac {\d x} {q + p \sec a x} = \frac x q - \frac p q \int \frac {\d x} {p + q \cos a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {q + p \sec a x}
| r = \frac 1 q \int \frac {q \rd x} {q + p \sec a x}
| c = multiplying top and bottom by $q$
}}
{{eqn | r = \frac 1 q \int \frac {\paren {q + p \sec a x - p \sec a x} \rd x} {q + p \sec a x}
| c =
}}
{{eqn | r = \frac 1 q \int \frac {\pare... | :$\ds \int \frac {\d x} {q + p \sec a x} = \frac x q - \frac p q \int \frac {\d x} {p + q \cos a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {q + p \sec a x}
| r = \frac 1 q \int \frac {q \rd x} {q + p \sec a x}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $q$
}}
{{eqn | r = \frac 1 q \int \frac {\paren {q + p \sec a x - p \sec a x} \rd x} {q + p \sec a x}
... | Primitive of Reciprocal of q plus p by Secant of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_q_plus_p_by_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_q_plus_p_by_Secant_of_a_x | [
"Primitives involving Secant Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Secant is Reciprocal of Cosine"
] |
proofwiki-9654 | Primitive of Cube of Cosecant of a x | :$\ds \int \csc^3 a x \rd x = \frac {-\csc a x \cot a x} {2 a} + \frac 1 {2 a} \ln \size {\tan \dfrac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \csc^3 x \rd x
| r = \frac {-\csc a x \cot a x} {2 a} + \frac 1 2 \int \csc a x \rd x
| c = Primitive of $\csc^n a x$ where $n = 3$
}}
{{eqn | r = \frac {-\csc a x \cot a x} {2 a} + \frac 1 2 \paren {\frac 1 a \ln \size {\tan \dfrac {a x} 2} }
| c = Primitive of $\csc a ... | :$\ds \int \csc^3 a x \rd x = \frac {-\csc a x \cot a x} {2 a} + \frac 1 {2 a} \ln \size {\tan \dfrac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \csc^3 x \rd x
| r = \frac {-\csc a x \cot a x} {2 a} + \frac 1 2 \int \csc a x \rd x
| c = [[Primitive of Power of Cosecant of a x|Primitive of $\csc^n a x$]] where $n = 3$
}}
{{eqn | r = \frac {-\csc a x \cot a x} {2 a} + \frac 1 2 \paren {\frac 1 a \ln \size {\tan \dfrac {a... | Primitive of Cube of Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Cosecant_of_a_x | [
"Primitives involving Cosecant Function"
] | [] | [
"Primitive of Power of Cosecant of a x",
"Primitive of Cosecant of a x/Tangent Form"
] |
proofwiki-9655 | Primitive of Reciprocal of Cosecant of a x | :$\ds \int \frac {\d x} {\csc a x} = \frac {-\cos a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\csc a x}
| r = \int \sin a x \rd x
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac {-\cos a x} a + C
| c = Primitive of $\sin a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {\csc a x} = \frac {-\cos a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\csc a x}
| r = \int \sin a x \rd x
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | r = \frac {-\cos a x} a + C
| c = [[Primitive of Sine of a x|Primitive of $\sin a x$]]
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cosecant_of_a_x | [
"Primitives involving Cosecant Function"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Primitive of Sine Function/Corollary"
] |
proofwiki-9656 | Primitive of x by Cosecant of a x | :$\ds \int x \csc a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} {18} + \frac {7 \paren {a x}^5} {1800} + \cdots + \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
where $B_{2 n}$ is the $2 n$th Bernoulli number. | {{begin-eqn}}
{{eqn | l = \int x \csc a x \rd x
| r = \frac 1 {a^2} \int \theta \csc \theta \rd \theta
| c = Substitution of $a x \to \theta$
}}
{{eqn | r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} \rd... | :$\ds \int x \csc a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} {18} + \frac {7 \paren {a x}^5} {1800} + \cdots + \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
where $B_{2 n}$ is the $2 n$th [[Definition:Bernoulli Numbers|Berno... | {{begin-eqn}}
{{eqn | l = \int x \csc a x \rd x
| r = \frac 1 {a^2} \int \theta \csc \theta \rd \theta
| c = [[Integration by Substitution|Substitution of $a x \to \theta$]]
}}
{{eqn | r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \, \theta... | Primitive of x by Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Cosecant_of_a_x | [
"Primitives involving Cosecant Function"
] | [
"Definition:Bernoulli Numbers"
] | [
"Integration by Substitution",
"Power Series Expansion for Cosecant Function",
"Power Series is Termwise Integrable within Radius of Convergence",
"Integration by Substitution"
] |
proofwiki-9657 | Primitive of Cosecant of a x over x | :$\ds \int \frac {\csc a x} x \rd x = \frac {-1} {a x} + \frac {a x} 6 + \frac {7 \paren {a x}^3} {1080} + \cdots + \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + \cdots + C$
where $B_n$ is the $n$th Bernoulli number. | {{begin-eqn}}
{{eqn | l = \int \frac {\csc a x} x \rd x
| r = \int \frac 1 x \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n}!} \rd x
| c = Power Series Expansion for Cosecant Function
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty ... | :$\ds \int \frac {\csc a x} x \rd x = \frac {-1} {a x} + \frac {a x} 6 + \frac {7 \paren {a x}^3} {1080} + \cdots + \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + \cdots + C$
where $B_n$ is the $n$th [[Definition:Bernoulli Numbers|Bernoulli num... | {{begin-eqn}}
{{eqn | l = \int \frac {\csc a x} x \rd x
| r = \int \frac 1 x \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n}!} \rd x
| c = [[Power Series Expansion for Cosecant Function]]
}}
{{eqn | r = \sum_{n \mathop = 0}^\in... | Primitive of Cosecant of a x over x | https://proofwiki.org/wiki/Primitive_of_Cosecant_of_a_x_over_x | https://proofwiki.org/wiki/Primitive_of_Cosecant_of_a_x_over_x | [
"Primitives involving Cosecant Function"
] | [
"Definition:Bernoulli Numbers"
] | [
"Power Series Expansion for Cosecant Function",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9658 | Primitive of x by Square of Cosecant of a x | :$\ds \int x \csc^2 a x \rd x = \frac {-x \cot a x} a + \frac 1 {a^2} \ln \size {\sin a x} + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int x \csc^2 a x \rd x = \frac {-x \cot a x} a + \frac 1 {a^2} \ln \size {\sin a x} + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x by Square of Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Cosecant_of_a_x | [
"Primitives involving Cosecant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Square of Cosecant of a x",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of Cotangent of a x"
] |
proofwiki-9659 | Primitive of Reciprocal of q plus p by Cosecant of a x | :$\ds \int \frac {\d x} {q + p \csc a x} = \frac x q - \frac p q \int \frac {\d x} {p + q \sin a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {q + p \csc a x}
| r = \frac 1 q \int \frac {q \rd x} {q + p \csc a x}
| c = multiplying top and bottom by $q$
}}
{{eqn | r = \frac 1 q \int \frac {\paren {q + p \csc a x - p \csc a x} \rd x} {q + p \csc a x}
| c =
}}
{{eqn | r = \frac 1 q \int \frac {\pare... | :$\ds \int \frac {\d x} {q + p \csc a x} = \frac x q - \frac p q \int \frac {\d x} {p + q \sin a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {q + p \csc a x}
| r = \frac 1 q \int \frac {q \rd x} {q + p \csc a x}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $q$
}}
{{eqn | r = \frac 1 q \int \frac {\paren {q + p \csc a x - p \csc a x} \rd x} {q + p \csc a x}
... | Primitive of Reciprocal of q plus p by Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_q_plus_p_by_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_q_plus_p_by_Cosecant_of_a_x | [
"Primitives involving Cosecant Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Cosecant is Reciprocal of Sine"
] |
proofwiki-9660 | Primitive of Arcsine of x over a | :$\ds \int \arcsin \frac x a \rd x = x \arcsin \frac x a + \sqrt {a^2 - x^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \arcsin x \rd x
| r = x \arcsin x + \sqrt {1 - x^2} + C
| c = Primitive of $\arcsin x$
}}
{{eqn | ll= \leadsto
| l = \int \arcsin \frac x a \rd x
| r = \frac 1 {1 / a} \paren {\frac x a \arcsin \frac x a + \sqrt {1 - \paren {\dfrac x a}^2} } + C
| c = Primiti... | :$\ds \int \arcsin \frac x a \rd x = x \arcsin \frac x a + \sqrt {a^2 - x^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \arcsin x \rd x
| r = x \arcsin x + \sqrt {1 - x^2} + C
| c = [[Primitive of Arcsine Function|Primitive of $\arcsin x$]]
}}
{{eqn | ll= \leadsto
| l = \int \arcsin \frac x a \rd x
| r = \frac 1 {1 / a} \paren {\frac x a \arcsin \frac x a + \sqrt {1 - \paren {\dfrac... | Primitive of Arcsine of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Arcsine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Arcsine_of_x_over_a/Proof_1 | [
"Primitive of Arcsine of x over a",
"Primitive of Arcsine Function",
"Primitives involving Inverse Sine Function"
] | [] | [
"Primitive of Arcsine Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9661 | Primitive of Arcsine of x over a | :$\ds \int \arcsin \frac x a \rd x = x \arcsin \frac x a + \sqrt {a^2 - x^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \sin u
| r = \frac x a
| c = {{Defof|Real Arcsine}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \cos u
| r = \sqrt {1 - \frac {x^2} {a^2} }
| c = Sum of Squares o... | :$\ds \int \arcsin \frac x a \rd x = x \arcsin \frac x a + \sqrt {a^2 - x^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \sin u
| r = \frac x a
| c = {{Defof|Real Arcsine}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \cos u
| r = \sqrt {1 - \frac {x^2} {a^2} }
| c = [[Sum of Squares... | Primitive of Arcsine of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Arcsine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Arcsine_of_x_over_a/Proof_2 | [
"Primitive of Arcsine of x over a",
"Primitive of Arcsine Function",
"Primitives involving Inverse Sine Function"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Primitive of Function of Arcsine",
"Primitive of x by Cosine of a x"
] |
proofwiki-9662 | Primitive of Arcsine of x over a | :$\ds \int \arcsin \frac x a \rd x = x \arcsin \frac x a + \sqrt {a^2 - x^2} + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {a^2 - x^2} }
| c = D... | :$\ds \int \arcsin \frac x a \rd x = x \arcsin \frac x a + \sqrt {a^2 - x^2} + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt... | Primitive of Arcsine of x over a/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Arcsine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Arcsine_of_x_over_a/Proof_3 | [
"Primitive of Arcsine of x over a",
"Primitive of Arcsine Function",
"Primitives involving Inverse Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arcsine Function/Corollary",
"Primitive of Constant",
"Integration by Parts",
"Primitive of x over Root of a squared minus x squared"
] |
proofwiki-9663 | Primitive of x by Arcsine of x over a | :$\ds \int x \arcsin \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arcsin \frac x a + \frac {x \sqrt {a^2 - x^2} } 4 + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \sin u
| r = \frac x a
| c = {{Defof|Real Arcsine}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \cos u
| r = \sqrt {1 - \frac {x^2} {a^2} }
| c = Sum of Squares o... | :$\ds \int x \arcsin \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arcsin \frac x a + \frac {x \sqrt {a^2 - x^2} } 4 + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \sin u
| r = \frac x a
| c = {{Defof|Real Arcsine}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \cos u
| r = \sqrt {1 - \frac {x^2} {a^2} }
| c = [[Sum of Squares... | Primitive of x by Arcsine of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_by_Arcsine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_by_Arcsine_of_x_over_a/Proof_1 | [
"Primitives involving Inverse Sine Function",
"Primitive of x by Arcsine of x over a"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Primitive of Function of Arcsine",
"Double Angle Formulas/Sine",
"Primitive of Constant Multiple of Function",
"Primitive of x by Sine of a x",
"Double Angle Formulas/Sine",
"Double Angle Formulas/Cosine"
] |
proofwiki-9664 | Primitive of x by Arcsine of x over a | :$\ds \int x \arcsin \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arcsin \frac x a + \frac {x \sqrt {a^2 - x^2} } 4 + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {a^2 - x^2} }
| c = D... | :$\ds \int x \arcsin \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arcsin \frac x a + \frac {x \sqrt {a^2 - x^2} } 4 + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {... | Primitive of x by Arcsine of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_by_Arcsine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_by_Arcsine_of_x_over_a/Proof_2 | [
"Primitives involving Inverse Sine Function",
"Primitive of x by Arcsine of x over a"
] | [] | [
"Definition:Primitive",
"Derivative of Arcsine Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x squared over Root of a squared minus x squared"
] |
proofwiki-9665 | Primitive of x squared by Arcsine of x over a | :$\ds \int x^2 \arcsin \frac x a \rd x = \frac {x^3} 3 \arcsin \frac x a + \frac {\paren {x^2 + 2 a^2} \sqrt {a^2 - x^2} } 9 + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {a^2 - x^2} }
| c = D... | :$\ds \int x^2 \arcsin \frac x a \rd x = \frac {x^3} 3 \arcsin \frac x a + \frac {\paren {x^2 + 2 a^2} \sqrt {a^2 - x^2} } 9 + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {... | Primitive of x squared by Arcsine of x over a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Arcsine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Arcsine_of_x_over_a | [
"Primitives involving Inverse Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arcsine Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x cubed over Root of a squared minus x squared"
] |
proofwiki-9666 | Primitive of Arcsine of x over a over x squared | :$\ds \int \dfrac 1 {x^2} \arcsin \frac x a \rd x = -\frac 1 x \arcsin \frac x a - \frac 1 a \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {a^2 - x^2} }
| c = D... | :$\ds \int \dfrac 1 {x^2} \arcsin \frac x a \rd x = -\frac 1 x \arcsin \frac x a - \frac 1 a \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {... | Primitive of Arcsine of x over a over x squared | https://proofwiki.org/wiki/Primitive_of_Arcsine_of_x_over_a_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Arcsine_of_x_over_a_over_x_squared | [
"Primitives involving Inverse Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arcsine Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Reciprocal of x by Root of a squared minus x squared/Logarithm Form"
] |
proofwiki-9667 | Primitive of Square of Arcsine of x over a | :$\ds \int \paren {\arcsin \frac x a}^2 \rd x = x \paren {\arcsin \frac x a}^2 - 2 x + 2 \sqrt{a^2 - x^2} \arcsin \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \sin u
| r = \frac x a
| c = {{Defof|Real Arcsine}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \cos u
| r = \sqrt {1 - \frac {x^2} {a^2} }
| c = Sum of Squares o... | :$\ds \int \paren {\arcsin \frac x a}^2 \rd x = x \paren {\arcsin \frac x a}^2 - 2 x + 2 \sqrt{a^2 - x^2} \arcsin \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \sin u
| r = \frac x a
| c = {{Defof|Real Arcsine}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \cos u
| r = \sqrt {1 - \frac {x^2} {a^2} }
| c = [[Sum of Squares... | Primitive of Square of Arcsine of x over a | https://proofwiki.org/wiki/Primitive_of_Square_of_Arcsine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Square_of_Arcsine_of_x_over_a | [
"Primitives involving Inverse Sine Function"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Primitive of Function of Arcsine",
"Primitive of x squared by Cosine of a x"
] |
proofwiki-9668 | Primitive of Arccosine of x over a | :$\ds \int \arccos \frac x a \rd x = x \arccos \frac x a - \sqrt {a^2 - x^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \cos u
| r = \frac x a
| c = {{Defof|Real Arccosine}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \cos u
| r = \sqrt {1 - \frac {x^2} {a^2} }
| c = Sum of Squares... | :$\ds \int \arccos \frac x a \rd x = x \arccos \frac x a - \sqrt {a^2 - x^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \cos u
| r = \frac x a
| c = {{Defof|Real Arccosine}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \cos u
| r = \sqrt {1 - \frac {x^2} {a^2} }
| c = [[Sum of Squar... | Primitive of Arccosine of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Arccosine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Arccosine_of_x_over_a/Proof_1 | [
"Primitive of Arccosine of x over a",
"Primitive of Arccosine Function",
"Primitives involving Inverse Cosine Function"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Primitive of Function of Arccosine",
"Primitive of x by Sine of a x"
] |
proofwiki-9669 | Primitive of Arccosine of x over a | :$\ds \int \arccos \frac x a \rd x = x \arccos \frac x a - \sqrt {a^2 - x^2} + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqrt {a^2 - x^2} }
| c ... | :$\ds \int \arccos \frac x a \rd x = x \arccos \frac x a - \sqrt {a^2 - x^2} + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqr... | Primitive of Arccosine of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Arccosine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Arccosine_of_x_over_a/Proof_2 | [
"Primitive of Arccosine of x over a",
"Primitive of Arccosine Function",
"Primitives involving Inverse Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccosine Function/Corollary",
"Primitive of Constant",
"Integration by Parts",
"Primitive of x over Root of a squared minus x squared"
] |
proofwiki-9670 | Primitive of Arccosine of x over a | :$\ds \int \arccos \frac x a \rd x = x \arccos \frac x a - \sqrt {a^2 - x^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \arccos x \rd x
| r = x \arccos x - \sqrt {1 - x^2} + C
| c = Primitive of $\arccos x$
}}
{{eqn | ll= \leadsto
| l = \int \arccos \frac x a \rd x
| r = \frac 1 {1 / a} \paren {\frac x a \arccos \frac x a - \sqrt {1 - \paren {\dfrac x a}^2} } + C
| c = Primiti... | :$\ds \int \arccos \frac x a \rd x = x \arccos \frac x a - \sqrt {a^2 - x^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \arccos x \rd x
| r = x \arccos x - \sqrt {1 - x^2} + C
| c = [[Primitive of Arccosine Function|Primitive of $\arccos x$]]
}}
{{eqn | ll= \leadsto
| l = \int \arccos \frac x a \rd x
| r = \frac 1 {1 / a} \paren {\frac x a \arccos \frac x a - \sqrt {1 - \paren {\dfr... | Primitive of Arccosine of x over a/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Arccosine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Arccosine_of_x_over_a/Proof_3 | [
"Primitive of Arccosine of x over a",
"Primitive of Arccosine Function",
"Primitives involving Inverse Cosine Function"
] | [] | [
"Primitive of Arccosine Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9671 | Primitive of x by Arccosine of x over a | :$\ds \int x \arccos \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arccos \frac x a - \frac {x \sqrt {a^2 - x^2} } 4 + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \cos u
| r = \frac x a
| c = {{Defof|Real Arccosine}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \sin u
| r = \sqrt {1 - \frac {x^2} {a^2} }
| c = Sum of Squares... | :$\ds \int x \arccos \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arccos \frac x a - \frac {x \sqrt {a^2 - x^2} } 4 + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \cos u
| r = \frac x a
| c = {{Defof|Real Arccosine}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \sin u
| r = \sqrt {1 - \frac {x^2} {a^2} }
| c = [[Sum of Squar... | Primitive of x by Arccosine of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_by_Arccosine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_by_Arccosine_of_x_over_a/Proof_1 | [
"Primitives involving Inverse Cosine Function",
"Primitive of x by Arccosine of x over a"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Primitive of Function of Arcsine",
"Double Angle Formulas/Sine",
"Primitive of Constant Multiple of Function",
"Primitive of x by Sine of a x",
"Double Angle Formulas/Sine",
"Double Angle Formulas/Cosine"
] |
proofwiki-9672 | Primitive of x by Arccosine of x over a | :$\ds \int x \arccos \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arccos \frac x a - \frac {x \sqrt {a^2 - x^2} } 4 + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqrt {a^2 - x^2} }
| c ... | :$\ds \int x \arccos \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arccos \frac x a - \frac {x \sqrt {a^2 - x^2} } 4 + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqr... | Primitive of x by Arccosine of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_by_Arccosine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_by_Arccosine_of_x_over_a/Proof_2 | [
"Primitives involving Inverse Cosine Function",
"Primitive of x by Arccosine of x over a"
] | [] | [
"Definition:Primitive",
"Derivative of Arccosine Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x squared over Root of a squared minus x squared",
"Sum of Arcsine and Arccosine",
"Definition:Primitive (Calculus)/Constant o... |
proofwiki-9673 | Primitive of x squared by Arccosine of x over a | :$\ds \int x^2 \arccos \frac x a \rd x = \frac {x^3} 3 \arccos \frac x a - \frac {\paren {x^2 + 2 a^2} \sqrt {a^2 - x^2} } 9 + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqrt {a^2 - x^2} }
| c ... | :$\ds \int x^2 \arccos \frac x a \rd x = \frac {x^3} 3 \arccos \frac x a - \frac {\paren {x^2 + 2 a^2} \sqrt {a^2 - x^2} } 9 + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqr... | Primitive of x squared by Arccosine of x over a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Arccosine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Arccosine_of_x_over_a | [
"Primitives involving Inverse Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccosine Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x cubed over Root of a squared minus x squared"
] |
proofwiki-9674 | Primitive of Arccosine of x over a over x | :$\ds \int \dfrac 1 x \arccos \frac x a \rd x = \frac \pi 2 \ln \size x - \int \frac {\arcsin \frac x a \rd x} x + C$ | {{begin-eqn}}
{{eqn | l = \int \dfrac 1 x \arccos \frac x a \rd x
| r = \int \dfrac 1 x \paren {\frac \pi 2 - \arcsin \frac x a} \rd x
| c = Sum of Arcsine and Arccosine
}}
{{eqn | r = \frac \pi 2 \int \frac {\d x} x - \int \dfrac 1 x \arcsin \frac x a \rd x
| c = Linear Combination of Primitives
}}
{... | :$\ds \int \dfrac 1 x \arccos \frac x a \rd x = \frac \pi 2 \ln \size x - \int \frac {\arcsin \frac x a \rd x} x + C$ | {{begin-eqn}}
{{eqn | l = \int \dfrac 1 x \arccos \frac x a \rd x
| r = \int \dfrac 1 x \paren {\frac \pi 2 - \arcsin \frac x a} \rd x
| c = [[Sum of Arcsine and Arccosine]]
}}
{{eqn | r = \frac \pi 2 \int \frac {\d x} x - \int \dfrac 1 x \arcsin \frac x a \rd x
| c = [[Linear Combination of Primitive... | Primitive of Arccosine of x over a over x | https://proofwiki.org/wiki/Primitive_of_Arccosine_of_x_over_a_over_x | https://proofwiki.org/wiki/Primitive_of_Arccosine_of_x_over_a_over_x | [
"Primitives involving Inverse Cosine Function"
] | [] | [
"Sum of Arcsine and Arccosine",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal"
] |
proofwiki-9675 | Primitive of Arccosine of x over a over x squared | :$\ds \int \frac 1 {x^2} \arccos \frac x a \rd x = -\frac 1 x \arccos \frac x a + \frac 1 a \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqrt {a^2 - x^2} }
| c ... | :$\ds \int \frac 1 {x^2} \arccos \frac x a \rd x = -\frac 1 x \arccos \frac x a + \frac 1 a \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqr... | Primitive of Arccosine of x over a over x squared | https://proofwiki.org/wiki/Primitive_of_Arccosine_of_x_over_a_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Arccosine_of_x_over_a_over_x_squared | [
"Primitives involving Inverse Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccosine Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Reciprocal of x by Root of a squared minus x squared/Logarithm Form",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Inverse Cosine/Real/Arccosine",
"Definition:Posi... |
proofwiki-9676 | Primitive of Square of Arccosine of x over a | :$\ds \int \paren {\arccos \frac x a}^2 \rd x = x \paren {\arccos \frac x a}^2 - 2 x - 2 \sqrt {a^2 - x^2} \arccos \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \cos u
| r = \frac x a
| c = {{Defof|Real Arccosine}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \sin u
| r = \sqrt {1 - \frac {x^2} {a^2} }
| c = Sum of Squares... | :$\ds \int \paren {\arccos \frac x a}^2 \rd x = x \paren {\arccos \frac x a}^2 - 2 x - 2 \sqrt {a^2 - x^2} \arccos \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \cos u
| r = \frac x a
| c = {{Defof|Real Arccosine}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \sin u
| r = \sqrt {1 - \frac {x^2} {a^2} }
| c = [[Sum of Squar... | Primitive of Square of Arccosine of x over a | https://proofwiki.org/wiki/Primitive_of_Square_of_Arccosine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Square_of_Arccosine_of_x_over_a | [
"Primitives involving Inverse Cosine Function"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Primitive of Function of Arccosine",
"Primitive of x squared by Sine of a x"
] |
proofwiki-9677 | Primitive of Arctangent of x over a | :$\ds \int \arctan \frac x a \rd x = x \arctan \frac x a - \frac a 2 \map \ln {x^2 + a^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \tan u
| r = \frac x a
| c = {{Defof|Real Arctangent}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \sec u
| r = \sqrt {1 + \frac {x^2} {a^2} }
| c = Difference of... | :$\ds \int \arctan \frac x a \rd x = x \arctan \frac x a - \frac a 2 \map \ln {x^2 + a^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \tan u
| r = \frac x a
| c = {{Defof|Real Arctangent}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \sec u
| r = \sqrt {1 + \frac {x^2} {a^2} }
| c = [[Difference ... | Primitive of Arctangent of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Arctangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Arctangent_of_x_over_a/Proof_1 | [
"Primitive of Arctangent of x over a",
"Primitive of Arctangent Function",
"Primitives involving Inverse Tangent Function"
] | [] | [
"Sum of Squares of Sine and Cosine/Corollary 1",
"Primitive of Function of Arctangent",
"Primitive of x by Square of Secant of a x",
"Logarithm of Reciprocal",
"Secant is Reciprocal of Cosine",
"Logarithm of Power",
"Difference of Logarithms",
"Definition:Primitive (Calculus)/Constant of Integration",... |
proofwiki-9678 | Primitive of Arctangent of x over a | :$\ds \int \arctan \frac x a \rd x = x \arctan \frac x a - \frac a 2 \map \ln {x^2 + a^2} + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {x^2 + a^2}
| c = Derivative... | :$\ds \int \arctan \frac x a \rd x = x \arctan \frac x a - \frac a 2 \map \ln {x^2 + a^2} + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {x^2 + a... | Primitive of Arctangent of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Arctangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Arctangent_of_x_over_a/Proof_2 | [
"Primitive of Arctangent of x over a",
"Primitive of Arctangent Function",
"Primitives involving Inverse Tangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arctangent Function/Corollary",
"Primitive of Constant",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x over x squared plus a squared"
] |
proofwiki-9679 | Primitive of Arctangent of x over a | :$\ds \int \arctan \frac x a \rd x = x \arctan \frac x a - \frac a 2 \map \ln {x^2 + a^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \arctan x \rd x
| r = x \arctan x - \frac {\map \ln {x^2 + 1} } 2 + C
| c = Primitive of $\arctan x$
}}
{{eqn | ll= \leadsto
| l = \int \arctan \frac x a \rd x
| r = \frac 1 {1 / a} \paren {\frac x a \arctan \frac x a - \dfrac 1 2 \map \ln {\paren {\frac x a}^2 + 1... | :$\ds \int \arctan \frac x a \rd x = x \arctan \frac x a - \frac a 2 \map \ln {x^2 + a^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \arctan x \rd x
| r = x \arctan x - \frac {\map \ln {x^2 + 1} } 2 + C
| c = [[Primitive of Arctangent Function|Primitive of $\arctan x$]]
}}
{{eqn | ll= \leadsto
| l = \int \arctan \frac x a \rd x
| r = \frac 1 {1 / a} \paren {\frac x a \arctan \frac x a - \dfrac 1... | Primitive of Arctangent of x over a/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Arctangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Arctangent_of_x_over_a/Proof_3 | [
"Primitive of Arctangent of x over a",
"Primitive of Arctangent Function",
"Primitives involving Inverse Tangent Function"
] | [] | [
"Primitive of Arctangent Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9680 | Primitive of x by Arctangent of x over a | :$\ds \int x \arctan \frac x a \rd x = \frac {x^2 + a^2} 2 \arctan \frac x a - \frac {a x} 2 + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {x^2 + a^2}
| c = Derivative... | :$\ds \int x \arctan \frac x a \rd x = \frac {x^2 + a^2} 2 \arctan \frac x a - \frac {a x} 2 + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {x^2 + a... | Primitive of x by Arctangent of x over a | https://proofwiki.org/wiki/Primitive_of_x_by_Arctangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_by_Arctangent_of_x_over_a | [
"Primitives involving Inverse Tangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arctangent Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x squared over x squared plus a squared"
] |
proofwiki-9681 | Primitive of x by Arccotangent of x over a | :$\ds \int x \arccot \frac x a \rd x = \frac {x^2 + a^2} 2 \arccot \frac x a + \frac {a x} 2 + C$ | <onlyinclude>
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 + a^2}
... | :$\ds \int x \arccot \frac x a \rd x = \frac {x^2 + a^2} 2 \arccot \frac x a + \frac {a x} 2 + C$ | <onlyinclude>
With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \f... | Primitive of x by Arccotangent of x over a | https://proofwiki.org/wiki/Primitive_of_x_by_Arccotangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_by_Arccotangent_of_x_over_a | [
"Primitives involving Inverse Cotangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccotangent Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x squared over x squared plus a squared",
"Sum of Arctangent and Arccotangent",
"Definition:Primitive (Calculus)/Constant o... |
proofwiki-9682 | Primitive of x squared by Arctangent of x over a | :$\ds \int x^2 \arctan \frac x a \rd x = \frac {x^3} 3 \arctan \frac x a - \frac {a x^2} 6 + \frac {a^3} 6 \map \ln {x^2 + a^2} + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {x^2 + a^2}
| c = Derivative... | :$\ds \int x^2 \arctan \frac x a \rd x = \frac {x^3} 3 \arctan \frac x a - \frac {a x^2} 6 + \frac {a^3} 6 \map \ln {x^2 + a^2} + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {x^2 + a... | Primitive of x squared by Arctangent of x over a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Arctangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Arctangent_of_x_over_a | [
"Primitives involving Inverse Tangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arctangent Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x cubed over x squared plus a squared"
] |
proofwiki-9683 | Primitive of x squared by Arccotangent of x over a | :$\ds \int x^2 \arccot \frac x a \rd x = \frac {x^3} 3 \arccot \frac x a + \frac {a x^2} 6 - \frac {a^3} 6 \map \ln {x^2 + a^2} + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 + a^2}
| c = Derivat... | :$\ds \int x^2 \arccot \frac x a \rd x = \frac {x^3} 3 \arccot \frac x a + \frac {a x^2} 6 - \frac {a^3} 6 \map \ln {x^2 + a^2} + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 ... | Primitive of x squared by Arccotangent of x over a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Arccotangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Arccotangent_of_x_over_a | [
"Primitives involving Inverse Cotangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccotangent Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x cubed over x squared plus a squared"
] |
proofwiki-9684 | Primitive of Arccotangent of x over a over x | :$\ds \int \dfrac 1 x \arccot \frac x a \rd x = \frac \pi 2 \ln \size x - \int \dfrac 1 x \arctan \frac x a \rd x$ | {{begin-eqn}}
{{eqn | l = \int \dfrac 1 x \arccot \frac x a \rd x
| r = \int \dfrac 1 x \paren {\frac \pi 2 - \arctan \frac x a} \rd x
| c = Sum of Arctangent and Arccotangent
}}
{{eqn | r = \frac \pi 2 \int \frac {\d x} x - \int \dfrac 1 x \arctan \frac x a \rd x
| c = Linear Combination of Primitive... | :$\ds \int \dfrac 1 x \arccot \frac x a \rd x = \frac \pi 2 \ln \size x - \int \dfrac 1 x \arctan \frac x a \rd x$ | {{begin-eqn}}
{{eqn | l = \int \dfrac 1 x \arccot \frac x a \rd x
| r = \int \dfrac 1 x \paren {\frac \pi 2 - \arctan \frac x a} \rd x
| c = [[Sum of Arctangent and Arccotangent]]
}}
{{eqn | r = \frac \pi 2 \int \frac {\d x} x - \int \dfrac 1 x \arctan \frac x a \rd x
| c = [[Linear Combination of Pri... | Primitive of Arccotangent of x over a over x | https://proofwiki.org/wiki/Primitive_of_Arccotangent_of_x_over_a_over_x | https://proofwiki.org/wiki/Primitive_of_Arccotangent_of_x_over_a_over_x | [
"Primitives involving Inverse Cotangent Function"
] | [] | [
"Sum of Arctangent and Arccotangent",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal"
] |
proofwiki-9685 | Primitive of Arctangent of x over a over x squared | :$\ds \int \frac 1 {x^2} \arctan \frac x a \rd x = -\frac 1 x \arctan \frac x a - \frac 1 {2 a} \map \ln {\frac {x^2 + a^2} {x^2} } + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {x^2 + a^2}
| c = Derivative... | :$\ds \int \frac 1 {x^2} \arctan \frac x a \rd x = -\frac 1 x \arctan \frac x a - \frac 1 {2 a} \map \ln {\frac {x^2 + a^2} {x^2} } + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {x^2 + a... | Primitive of Arctangent of x over a over x squared | https://proofwiki.org/wiki/Primitive_of_Arctangent_of_x_over_a_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Arctangent_of_x_over_a_over_x_squared | [
"Primitives involving Inverse Tangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arctangent Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Reciprocal of x by x squared plus a squared",
"Logarithm of Reciprocal"
] |
proofwiki-9686 | Primitive of Arccotangent of x over a over x squared | :$\ds \int \frac 1 {x^2} \arccot \frac x a \rd x = -\frac 1 x \arccot \frac x a + \frac 1 {2 a} \map \ln {\frac {x^2 + a^2} {x^2} } + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 + a^2}
| c = Derivat... | :$\ds \int \frac 1 {x^2} \arccot \frac x a \rd x = -\frac 1 x \arccot \frac x a + \frac 1 {2 a} \map \ln {\frac {x^2 + a^2} {x^2} } + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2... | Primitive of Arccotangent of x over a over x squared | https://proofwiki.org/wiki/Primitive_of_Arccotangent_of_x_over_a_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Arccotangent_of_x_over_a_over_x_squared | [
"Primitives involving Inverse Cotangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccotangent Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Reciprocal of x by x squared plus a squared",
"Logarithm of Reciprocal"
] |
proofwiki-9687 | Primitive of Arccotangent of x over a | :$\ds \int \arccot \frac x a \rd x = x \arccot \frac x a + \frac a 2 \map \ln {x^2 + a^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \cot u
| r = \frac x a
| c = {{Defof|Arccotangent}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \csc u
| r = \sqrt {1 + \frac {x^2} {a^2} }
| c = Difference of Sq... | :$\ds \int \arccot \frac x a \rd x = x \arccot \frac x a + \frac a 2 \map \ln {x^2 + a^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \cot u
| r = \frac x a
| c = {{Defof|Arccotangent}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \csc u
| r = \sqrt {1 + \frac {x^2} {a^2} }
| c = [[Difference of ... | Primitive of Arccotangent of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Arccotangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Arccotangent_of_x_over_a/Proof_1 | [
"Primitive of Arccotangent of x over a",
"Primitive of Arccotangent Function",
"Primitives involving Inverse Cotangent Function"
] | [] | [
"Sum of Squares of Sine and Cosine/Corollary 2",
"Primitive of Function of Arccotangent",
"Primitive of x by Square of Cosecant of a x",
"Logarithm of Reciprocal",
"Cosecant is Reciprocal of Sine",
"Logarithm of Power",
"Difference of Logarithms",
"Definition:Primitive (Calculus)/Constant of Integrati... |
proofwiki-9688 | Primitive of Arccotangent of x over a | :$\ds \int \arccot \frac x a \rd x = x \arccot \frac x a + \frac a 2 \map \ln {x^2 + a^2} + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 + a^2}
| c = Derivat... | :$\ds \int \arccot \frac x a \rd x = x \arccot \frac x a + \frac a 2 \map \ln {x^2 + a^2} + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 ... | Primitive of Arccotangent of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Arccotangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Arccotangent_of_x_over_a/Proof_2 | [
"Primitive of Arccotangent of x over a",
"Primitive of Arccotangent Function",
"Primitives involving Inverse Cotangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccotangent Function/Corollary",
"Primitive of Constant",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x over x squared plus a squared"
] |
proofwiki-9689 | Primitive of Arccosecant of x over a | :<nowiki>$\ds \int \arccsc \frac x a \rd x = \begin {cases}
x \arccsc \dfrac x a + a \map \ln {x + \sqrt {x^2 - a^2} } + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\
x \arccsc \dfrac x a - a \map \ln {x + \sqrt {x^2 - a^2} } + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\
\end {cases}$
</nowiki> | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki>\begin{cases} \dfrac {-a} {x \sqrt... | :<nowiki>$\ds \int \arccsc \frac x a \rd x = \begin {cases}
x \arccsc \dfrac x a + a \map \ln {x + \sqrt {x^2 - a^2} } + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\
x \arccsc \dfrac x a - a \map \ln {x + \sqrt {x^2 - a^2} } + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\
\end {cases}$
</nowiki> | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki>\begin{c... | Primitive of Arccosecant of x over a | https://proofwiki.org/wiki/Primitive_of_Arccosecant_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Arccosecant_of_x_over_a | [
"Primitives involving Inverse Cosecant Function",
"Arccosecant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccosecant Function/Corollary",
"Primitive of Constant",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form",
"Definition:Real Interv... |
proofwiki-9690 | Primitive of x by Arcsecant of x over a | :$\ds \int x \arcsec \frac x a \rd x = \begin{cases}
\dfrac {x^2} 2 \arcsec \dfrac x a - \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^2} 2 \arcsec \dfrac x a + \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\
\end{cases}$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begin {cases} \dfrac a {x \sqrt {x^2 - a^... | :$\ds \int x \arcsec \frac x a \rd x = \begin{cases}
\dfrac {x^2} 2 \arcsec \dfrac x a - \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^2} 2 \arcsec \dfrac x a + \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\
\end{cases}$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begin {cases} \... | Primitive of x by Arcsecant of x over a | https://proofwiki.org/wiki/Primitive_of_x_by_Arcsecant_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_by_Arcsecant_of_x_over_a | [
"Primitives involving Inverse Secant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arcsecant Function/Corollary 1",
"Primitive of Power",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x over Root of x squared minus a squared",
"Definition:Real Interval/Open",
"Integration ... |
proofwiki-9691 | Primitive of x by Arccosecant of x over a | :<nowiki>$\ds \int x \arccsc \frac x a \rd x = \begin{cases}
\dfrac {x^2} 2 \arccsc \dfrac x a + \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^2} 2 \arccsc \dfrac x a - \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\
\end{cases}$</nowiki> | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki>\begin {cases} \dfrac {-a} {x \sqr... | :<nowiki>$\ds \int x \arccsc \frac x a \rd x = \begin{cases}
\dfrac {x^2} 2 \arccsc \dfrac x a + \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^2} 2 \arccsc \dfrac x a - \dfrac {a \sqrt {x^2 - a^2} } 2 + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\
\end{cases}$</nowiki> | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki>\begin {... | Primitive of x by Arccosecant of x over a | https://proofwiki.org/wiki/Primitive_of_x_by_Arccosecant_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_by_Arccosecant_of_x_over_a | [
"Primitives involving Inverse Cosecant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccosecant Function/Corollary",
"Primitive of Power",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x over Root of x squared minus a squared",
"Definition:Real Interval/Open",
"Integration ... |
proofwiki-9692 | Primitive of x squared by Arcsecant of x over a | :$\ds \int x^2 \arcsec \frac x a \rd x = \begin{cases}
\dfrac {x^3} 3 \arcsec \dfrac x a - \dfrac {a x \sqrt {x^2 - a^2} } 6 - \dfrac {a^3} 6 \map \ln {x + \sqrt {x^2 - a^2} } + C & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^3} 3 \arcsec \dfrac x a + \dfrac {a x \sqrt {x^2 - a^2} } 6 + \dfrac {a^3} 6 \map \ln... | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begin{cases} \dfrac a {x \sqrt {x^2 - a^2... | :$\ds \int x^2 \arcsec \frac x a \rd x = \begin{cases}
\dfrac {x^3} 3 \arcsec \dfrac x a - \dfrac {a x \sqrt {x^2 - a^2} } 6 - \dfrac {a^3} 6 \map \ln {x + \sqrt {x^2 - a^2} } + C & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^3} 3 \arcsec \dfrac x a + \dfrac {a x \sqrt {x^2 - a^2} } 6 + \dfrac {a^3} 6 \map \ln... | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begin{cases} \d... | Primitive of x squared by Arcsecant of x over a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Arcsecant_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Arcsecant_of_x_over_a | [
"Primitives involving Inverse Secant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arcsecant Function/Corollary 1",
"Primitive of Power",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x squared over Root of x squared minus a squared",
"Definition:Real Interval/Open",
"Inte... |
proofwiki-9693 | Primitive of x squared by Arccosecant of x over a | :$\ds \int x^2 \arccsc \frac x a \rd x = \begin {cases}
\dfrac {x^3} 3 \arccsc \dfrac x a - \dfrac {a x \sqrt{x^2 - a^2} } 6 - \dfrac {a^3} 6 \map \ln {x + \sqrt {x^2 - a^2} } + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^3} 3 \arccsc \dfrac x a + \dfrac {a x \sqrt{x^2 - a^2} } 6 + \dfrac {a^3} 6 \map \ln ... | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begin {cases} \dfrac {-a} {x \sqrt {x^2 -... | :$\ds \int x^2 \arccsc \frac x a \rd x = \begin {cases}
\dfrac {x^3} 3 \arccsc \dfrac x a - \dfrac {a x \sqrt{x^2 - a^2} } 6 - \dfrac {a^3} 6 \map \ln {x + \sqrt {x^2 - a^2} } + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\
\dfrac {x^3} 3 \arccsc \dfrac x a + \dfrac {a x \sqrt{x^2 - a^2} } 6 + \dfrac {a^3} 6 \map \ln ... | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begin {cases} \... | Primitive of x squared by Arccosecant of x over a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Arccosecant_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Arccosecant_of_x_over_a | [
"Primitives involving Inverse Cosecant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccosecant Function/Corollary",
"Primitive of Power",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x squared over Root of x squared minus a squared",
"Definition:Real Interval/Open",
"Inte... |
proofwiki-9694 | Primitive of Arcsecant of x over a over x | <onlyinclude>
{{begin-eqn}}
{{eqn | l = \int \dfrac 1 x \arcsec \frac x a \rd x
| r = \frac \pi 2 \ln \size x + \sum_{n \mathop \ge 0} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1}^2} \paren {\frac a x}^{2 n + 1} + C
| c =
}}
{{eqn | r = \frac \pi 2 \ln \size x + \frac a x + \frac 1 {2 \tim... | {{begin-eqn}}
{{eqn | l = \arcsec \frac x a
| r = \frac \pi 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } \paren {\frac a x}^{2 n + 1}
| c = Power Series Expansion for Real Arcsecant Function
}}
{{eqn | ll= \leadsto
| l = \frac 1 x \arcsec \frac x a
... | <onlyinclude>
{{begin-eqn}}
{{eqn | l = \int \dfrac 1 x \arcsec \frac x a \rd x
| r = \frac \pi 2 \ln \size x + \sum_{n \mathop \ge 0} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1}^2} \paren {\frac a x}^{2 n + 1} + C
| c =
}}
{{eqn | r = \frac \pi 2 \ln \size x + \frac a x + \frac 1 {2 \tim... | {{begin-eqn}}
{{eqn | l = \arcsec \frac x a
| r = \frac \pi 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } \paren {\frac a x}^{2 n + 1}
| c = [[Power Series Expansion for Real Arcsecant Function]]
}}
{{eqn | ll= \leadsto
| l = \frac 1 x \arcsec \frac x ... | Primitive of Arcsecant of x over a over x | https://proofwiki.org/wiki/Primitive_of_Arcsecant_of_x_over_a_over_x | https://proofwiki.org/wiki/Primitive_of_Arcsecant_of_x_over_a_over_x | [
"Primitives involving Inverse Secant Function"
] | [] | [
"Power Series Expansion for Real Arcsecant Function",
"Fubini's Theorem",
"Primitive of Reciprocal",
"Primitive of Power"
] |
proofwiki-9695 | Primitive of Arcsecant of x over a over x squared | :$\ds \int \dfrac 1 {x^2} \arcsec \frac x a \rd x = \begin {cases}
-\dfrac 1 x \arcsec \dfrac x a + \dfrac {\sqrt {x^2 - a^2} } {a x} + C & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
-\dfrac 1 x \arcsec \dfrac x a - \dfrac {\sqrt {x^2 - a^2} } {a x} + C & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\
\end {cases}$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begin {cases} \dfrac a {x \sqrt {x^2 - a^... | :$\ds \int \dfrac 1 {x^2} \arcsec \frac x a \rd x = \begin {cases}
-\dfrac 1 x \arcsec \dfrac x a + \dfrac {\sqrt {x^2 - a^2} } {a x} + C & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
-\dfrac 1 x \arcsec \dfrac x a - \dfrac {\sqrt {x^2 - a^2} } {a x} + C & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\
\end {cases}$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begin {cases} \... | Primitive of Arcsecant of x over a over x squared | https://proofwiki.org/wiki/Primitive_of_Arcsecant_of_x_over_a_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Arcsecant_of_x_over_a_over_x_squared | [
"Primitives involving Inverse Secant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arcsecant Function/Corollary 1",
"Primitive of Power",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of x squared by Root of x squared minus a squared",
"Definition:Real Interval/Op... |
proofwiki-9696 | Primitive of Arccosecant of x over a over x squared | :$\ds \int \dfrac 1 {x^2} \arccsc \frac x a \rd x = \begin{cases}
-\dfrac 1 x \arccsc \dfrac x a - \dfrac {\sqrt{x^2 - a^2} } {a x} + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\
-\dfrac 1 x \arccsc \dfrac x a + \dfrac {\sqrt{x^2 - a^2} } {a x} + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\
\end{cases}$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begin {cases} \dfrac {-a} {x \sqrt {x^2 -... | :$\ds \int \dfrac 1 {x^2} \arccsc \frac x a \rd x = \begin{cases}
-\dfrac 1 x \arccsc \dfrac x a - \dfrac {\sqrt{x^2 - a^2} } {a x} + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\
-\dfrac 1 x \arccsc \dfrac x a + \dfrac {\sqrt{x^2 - a^2} } {a x} + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\
\end{cases}$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begin {cases} \... | Primitive of Arccosecant of x over a over x squared | https://proofwiki.org/wiki/Primitive_of_Arccosecant_of_x_over_a_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Arccosecant_of_x_over_a_over_x_squared | [
"Primitives involving Inverse Cosecant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccosecant Function/Corollary",
"Primitive of Power",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of x squared by Root of x squared minus a squared",
"Definition:Real Interval/Op... |
proofwiki-9697 | Primitive of Power of x by Arcsine of x over a | :$\ds \int x^m \arcsin \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arcsin \frac x a - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {a^2 - x^2} }$ | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arcsin x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arcsin x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }
| c = Primitive of $x^m \arcsin x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^m \arcsin \frac x a \rd x
| r =... | :$\ds \int x^m \arcsin \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arcsin \frac x a - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {a^2 - x^2} }$ | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arcsin x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arcsin x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }
| c = [[Primitive of Power of x by Arcsine of x|Primitive of $x^m \arcsin x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l... | Primitive of Power of x by Arcsine of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsine_of_x_over_a/Proof_1 | [
"Primitive of Power of x by Arcsine of x over a",
"Primitives involving Inverse Sine Function"
] | [] | [
"Primitive of Power of x by Arcsine of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9698 | Primitive of Power of x by Arcsine of x over a | :$\ds \int x^m \arcsin \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arcsin \frac x a - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {a^2 - x^2} }$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {a^2 - x^2} }
| c = D... | :$\ds \int x^m \arcsin \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arcsin \frac x a - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {a^2 - x^2} }$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {... | Primitive of Power of x by Arcsine of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsine_of_x_over_a/Proof_2 | [
"Primitive of Power of x by Arcsine of x over a",
"Primitives involving Inverse Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arcsine Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9699 | Primitive of Power of x by Arccosine of x over a | :$\ds \int x^m \arccos \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arccos \frac x a + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {a^2 - x^2} }$ | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccos x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arccos x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }
| c = Primitive of $x^m \arccos x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^m \arccos \frac x a \rd x
| r =... | :$\ds \int x^m \arccos \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arccos \frac x a + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {a^2 - x^2} }$ | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccos x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arccos x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }
| c = [[Primitive of Power of x by Arccosine of x|Primitive of $x^m \arccos x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn |... | Primitive of Power of x by Arccosine of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosine_of_x_over_a/Proof_1 | [
"Primitive of Power of x by Arccosine of x over a",
"Primitives involving Inverse Cosine Function"
] | [] | [
"Primitive of Power of x by Arccosine of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.