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-9500 | Primitive of x by Sine of a x | :$\ds \int x \sin a x \rd x = \frac {\sin a x} {a^2} - \frac {x \cos a x} a + C$ | 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... | :$\ds \int x \sin a x \rd x = \frac {\sin a x} {a^2} - \frac {x \cos a x} a + C$ | 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 Sine of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Sine Function/Corollary",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Cosine Function/Corollary"
] |
proofwiki-9501 | Primitive of x squared by Sine of a x | :$\ds \int x^2 \sin a x \rd x = \frac {2 x \sin a x} {a^2} + \paren {\frac 2 {a^3} - \frac {x^2} a} \cos a x + C$ | 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^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = Derivative of Power
}}
{{end-eqn}}
and l... | :$\ds \int x^2 \sin a x \rd x = \frac {2 x \sin a x} {a^2} + \paren {\frac 2 {a^3} - \frac {x^2} a} \cos a x + C$ | 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^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = [[Derivative o... | Primitive of x squared by Sine of a x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Primitive of Sine Function/Corollary",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of x by Cosine of a x"
] |
proofwiki-9502 | Primitive of x cubed by Sine of a x | :$\ds \int x^3 \sin a x \rd x = \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \sin a x + \paren {\frac {6 x} {a^3} - \frac {x^3} a} \cos a x + C$ | 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^3
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 3 x^2
| c = Derivative of Power
}}
{{end-eqn}}
and... | :$\ds \int x^3 \sin a x \rd x = \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \sin a x + \paren {\frac {6 x} {a^3} - \frac {x^3} a} \cos a x + C$ | 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^3
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 3 x^2
| c = [[Derivative... | Primitive of x cubed by Sine of a x | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Primitive of Sine Function/Corollary",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of x squared by Cosine of a x"
] |
proofwiki-9503 | Primitive of Sine of a x over x squared | :$\ds \int \frac {\sin a x \rd x} {x^2} = -\frac {\sin a x} x + a \int \frac {\cos a x \rd x} x + C$ | 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 a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cos a x
| c = Derivative of $\sin a x$
}}
... | :$\ds \int \frac {\sin a x \rd x} {x^2} = -\frac {\sin a x} x + a \int \frac {\cos a x \rd x} x + C$ | 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 a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cos a x
| c = [[... | Primitive of Sine of a x over x squared | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_x_squared | [
"Primitives involving Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Sine Function/Corollary",
"Primitive of Power",
"Integration by Parts"
] |
proofwiki-9504 | Primitive of Reciprocal of Sine of a x/Logarithm of Cosecant minus Cotangent Form | :$\ds \int \frac {\d x} {\sin a x} = \frac 1 a \ln \size {\csc a x - \cot a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin x}
| r = \int \csc x \rd x
| c = {{Defof|Real Cosecant Function}}
}}
{{eqn | r = \ln \size {\csc x - \cot x} + C
| c = Primitive of $\csc x$: Cosecant minus Cotangent Form
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\sin a x}
| r = \fr... | :$\ds \int \frac {\d x} {\sin a x} = \frac 1 a \ln \size {\csc a x - \cot a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin x}
| r = \int \csc x \rd x
| c = {{Defof|Real Cosecant Function}}
}}
{{eqn | r = \ln \size {\csc x - \cot x} + C
| c = [[Primitive of Cosecant Function/Cosecant minus Cotangent Form|Primitive of $\csc x$: Cosecant minus Cotangent Form]]
}}
{{eqn | ll= ... | Primitive of Reciprocal of Sine of a x/Logarithm of Cosecant minus Cotangent Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x/Logarithm_of_Cosecant_minus_Cotangent_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x/Logarithm_of_Cosecant_minus_Cotangent_Form | [
"Primitive of Reciprocal of Sine of a x"
] | [] | [
"Primitive of Cosecant Function/Cosecant minus Cotangent Form",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9505 | Primitive of Reciprocal of Sine of a x/Logarithm of Tangent Form | :$\ds \int \frac {\d x} {\sin a x} = \frac 1 a \ln \size {\tan \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin x}
| r = \int \csc x \rd x
| c = {{Defof|Real Cosecant Function}}
}}
{{eqn | r = \ln \size {\tan \frac x 2} + C
| c = Primitive of $\csc x$: Tangent Form
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\sin a x}
| r = \frac 1 a \ln \size {... | :$\ds \int \frac {\d x} {\sin a x} = \frac 1 a \ln \size {\tan \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin x}
| r = \int \csc x \rd x
| c = {{Defof|Real Cosecant Function}}
}}
{{eqn | r = \ln \size {\tan \frac x 2} + C
| c = [[Primitive of Cosecant Function/Tangent Form|Primitive of $\csc x$: Tangent Form]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d... | Primitive of Reciprocal of Sine of a x/Logarithm of Tangent Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x/Logarithm_of_Tangent_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x/Logarithm_of_Tangent_Form | [
"Primitive of Reciprocal of Sine of a x"
] | [] | [
"Primitive of Cosecant Function/Tangent Form",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9506 | Primitive of x over Sine of a x | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\sin a x}
| r = \frac 1 {a^2} \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 + 1}!} + C
| c =
}}
{{eqn | r = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} {18} + \frac {7 \p... | {{begin-eqn}}
{{eqn | l = \csc x
| r = \sum_{n \mathop = 0}^\infty \dfrac {B_{2 n} \paren {-1}^{n - 1} x^{2 n - 1} 2 \paren {2^{2 n - 1} - 1} } {\paren {2 n}!}
| c = Power Series Expansion for Cosecant Function
}}
{{eqn | ll= \leadsto
| l = \dfrac x {\sin a x}
| r = x \sum_{n \mathop = 0}^\infty... | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\sin a x}
| r = \frac 1 {a^2} \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 + 1}!} + C
| c =
}}
{{eqn | r = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} {18} + \frac {7 \p... | {{begin-eqn}}
{{eqn | l = \csc x
| r = \sum_{n \mathop = 0}^\infty \dfrac {B_{2 n} \paren {-1}^{n - 1} x^{2 n - 1} 2 \paren {2^{2 n - 1} - 1} } {\paren {2 n}!}
| c = [[Power Series Expansion for Cosecant Function]]
}}
{{eqn | ll= \leadsto
| l = \dfrac x {\sin a x}
| r = x \sum_{n \mathop = 0}^\i... | Primitive of x over Sine of a x | https://proofwiki.org/wiki/Primitive_of_x_over_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_over_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Power Series Expansion for Cosecant Function",
"Cosecant is Reciprocal of Sine",
"Primitive of Power",
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9507 | Primitive of Square of Sine of a x | :$\ds \int \sin^2 a x \rd x = \frac x 2 - \frac {\sin 2 a x} {4 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \sin^2 x \rd x
| r = \frac x 2 - \frac {\sin 2 x} 4 + C
| c = Primitive of $\sin^2 x$
}}
{{eqn | ll= \leadsto
| l = \int \sin^2 a x \rd x
| r = \frac 1 a \paren {\frac {a x} 2 - \frac {\sin 2 a x} 4} + C
| c = Primitive of Function of Constant Multiple
}}
{{e... | :$\ds \int \sin^2 a x \rd x = \frac x 2 - \frac {\sin 2 a x} {4 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \sin^2 x \rd x
| r = \frac x 2 - \frac {\sin 2 x} 4 + C
| c = [[Primitive of Square of Sine Function|Primitive of $\sin^2 x$]]
}}
{{eqn | ll= \leadsto
| l = \int \sin^2 a x \rd x
| r = \frac 1 a \paren {\frac {a x} 2 - \frac {\sin 2 a x} 4} + C
| c = [[Primit... | Primitive of Square of Sine of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Primitive of Square of Sine Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9508 | Primitive of x by Square of Sine of a x | :$\ds \int x \sin^2 a x \rd x = \frac {x^2} 4 - \frac {x \sin 2 a x} {4 a} - \frac {\cos 2 a x} {8 a^2} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int x \sin^2 a x \rd x = \frac {x^2} 4 - \frac {x \sin 2 a x} {4 a} - \frac {\cos 2 a x} {8 a^2} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x by Square of Sine of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Square of Sine of a x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Sine Function/Corollary"
] |
proofwiki-9509 | Primitive of Fourth Power of Sine of a x | :$\ds \int \sin^4 a x \rd x = \frac {3 x} 8 - \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \sin^4 a x \rd x
| r = \int \paren {\frac {3 - 4 \cos 2 a x + \cos 4 a x} 8} \rd x
| c = Power Reduction Formula for $\sin^4$
}}
{{eqn | r = \frac 3 8 \int \rd x - \frac 1 2 \int \cos 2 a x \rd x + \frac 1 8 \int \cos 4 a x \rd x
| c = Linear Combination of Primitives
}}... | :$\ds \int \sin^4 a x \rd x = \frac {3 x} 8 - \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \sin^4 a x \rd x
| r = \int \paren {\frac {3 - 4 \cos 2 a x + \cos 4 a x} 8} \rd x
| c = [[Power Reduction Formula for 4th Power of Sine|Power Reduction Formula for $\sin^4$]]
}}
{{eqn | r = \frac 3 8 \int \rd x - \frac 1 2 \int \cos 2 a x \rd x + \frac 1 8 \int \cos 4 a x \rd... | Primitive of Fourth Power of Sine of a x | https://proofwiki.org/wiki/Primitive_of_Fourth_Power_of_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Fourth_Power_of_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Power Reduction Formulas/Sine to 4th",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Cosine Function/Corollary"
] |
proofwiki-9510 | Primitive of Square of Cosecant of a x | :$\ds \int \csc^2 a x \rd x = -\frac {\cot a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \csc^2 x \rd x
| r = -\cot x + C
| c = Primitive of $\csc^2 x$
}}
{{eqn | ll= \leadsto
| l = \int \csc^2 a x \rd x
| r = \frac 1 a \paren {-\cot a x} + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = -\frac {\cot a x} a + C
| c = simpl... | :$\ds \int \csc^2 a x \rd x = -\frac {\cot a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \csc^2 x \rd x
| r = -\cot x + C
| c = [[Primitive of Square of Cosecant Function|Primitive of $\csc^2 x$]]
}}
{{eqn | ll= \leadsto
| l = \int \csc^2 a x \rd x
| r = \frac 1 a \paren {-\cot a x} + C
| c = [[Primitive of Function of Constant Multiple]]
}}
{{eq... | Primitive of Square of Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosecant_of_a_x | [
"Primitives involving Cosecant Function"
] | [] | [
"Primitive of Square of Cosecant Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9511 | Primitive of Reciprocal of Square of Sine of a x | :$\ds \int \frac {\d x} {\sin^2 a x} = \frac {-\cot a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin^2 a x}
| r = \int \csc^2 a x \rd x
| c = {{Defof|Cosecant|subdef = Analysis}}
}}
{{eqn | r = \frac {-\cot a x} a + C
| c = Primitive of $\csc^2 a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {\sin^2 a x} = \frac {-\cot a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin^2 a x}
| r = \int \csc^2 a x \rd x
| c = {{Defof|Cosecant|subdef = Analysis}}
}}
{{eqn | r = \frac {-\cot a x} a + C
| c = [[Primitive of Square of Cosecant of a x|Primitive of $\csc^2 a x$]]
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of Square of Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Sine_of_a_x | [
"Primitive of Reciprocal of Square of Sine of a x",
"Primitives involving Sine Function"
] | [] | [
"Primitive of Square of Cosecant of a x"
] |
proofwiki-9512 | Primitive of Reciprocal of Cube of Sine of a x | :$\ds \int \frac {\d x} {\sin^3 a x} = \frac {-\cos a x} {2 a \sin^2 a x} + \frac 1 {2 a} \ln \size {\tan \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin^3 a x}
| r = \int \csc^3 a x \rd x
| c = Cosecant is $\dfrac 1 \sin$
}}
{{eqn | r = \frac{-\csc a x \cot a x} {2 a} + \frac 1 2 \int \csc a x \rd x
| c = Primitive of $\csc^n a x$
}}
{{eqn | r = \frac {-\cos a x} {2 a \sin^2 a x} + \frac 1 2 \int \csc ... | :$\ds \int \frac {\d x} {\sin^3 a x} = \frac {-\cos a x} {2 a \sin^2 a x} + \frac 1 {2 a} \ln \size {\tan \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin^3 a x}
| r = \int \csc^3 a x \rd x
| c = [[Cosecant is Reciprocal of Sine|Cosecant is $\dfrac 1 \sin$]]
}}
{{eqn | 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 ... | Primitive of Reciprocal of Cube of Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cube_of_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cube_of_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Primitive of Power of Cosecant of a x",
"Cosecant is Reciprocal of Sine",
"Cotangent is Cosine divided by Sine",
"Primitive of Cosecant of a x/Tangent Form"
] |
proofwiki-9513 | Primitive of Sine of a x by Sine of b x | For $p \ne q$:
:$\ds \int \sin a x \sin b x \rd x = \frac {\map \sin {a - b} x} {2 \paren {a - b} } - \frac {\map \sin {a + b} x} {2 \paren {a + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \sin a x \sin b x \rd x
| r = \int \paren {\frac {\map \cos {a x - b x} - \map \cos {a x + b x} } 2} \rd x
| c = Werner Formula for Sine by Sine
}}
{{eqn | r = \frac 1 2 \int \map \cos {a - b} x \rd x - \frac 1 2 \int \map \cos {a + b} x \rd x
| c = Linear Combination of... | For $p \ne q$:
:$\ds \int \sin a x \sin b x \rd x = \frac {\map \sin {a - b} x} {2 \paren {a - b} } - \frac {\map \sin {a + b} x} {2 \paren {a + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \sin a x \sin b x \rd x
| r = \int \paren {\frac {\map \cos {a x - b x} - \map \cos {a x + b x} } 2} \rd x
| c = [[Werner Formula for Sine by Sine]]
}}
{{eqn | r = \frac 1 2 \int \map \cos {a - b} x \rd x - \frac 1 2 \int \map \cos {a + b} x \rd x
| c = [[Linear Combinat... | Primitive of Sine of a x by Sine of b x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_by_Sine_of_b_x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_by_Sine_of_b_x | [
"Primitives involving Sine Function"
] | [] | [
"Werner Formulas/Sine by Sine",
"Linear Combination of Integrals/Indefinite",
"Primitive of Cosine Function/Corollary"
] |
proofwiki-9514 | Primitive of Reciprocal of 1 minus Sine of a x | :$\ds \int \frac {\d x} {1 - \sin a x} = \frac 1 a \map \tan {\frac \pi 4 + \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {1 - \sin a x}
| r = \int \map {\sec^2} {\frac \pi 4 + \frac {a x} 2} \rd x
| c = Reciprocal of One Minus Sine
}}
{{eqn | n = 1
| r = \frac 1 2 \int \map {\sec^2} {\frac \pi 4 + \frac {a x} 2} \rd x
| c = Primitive of Constant Multiple of Function
}}
{... | :$\ds \int \frac {\d x} {1 - \sin a x} = \frac 1 a \map \tan {\frac \pi 4 + \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {1 - \sin a x}
| r = \int \map {\sec^2} {\frac \pi 4 + \frac {a x} 2} \rd x
| c = [[Reciprocal of One Minus Sine]]
}}
{{eqn | n = 1
| r = \frac 1 2 \int \map {\sec^2} {\frac \pi 4 + \frac {a x} 2} \rd x
| c = [[Primitive of Constant Multiple of Functio... | Primitive of Reciprocal of 1 minus Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_minus_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_minus_Sine_of_a_x | [
"Primitive of Reciprocal of 1 minus Sine of a x",
"Primitives involving Sine Function"
] | [] | [
"Reciprocal of One Minus Sine",
"Primitive of Constant Multiple of Function",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Square of Secant Function"
] |
proofwiki-9515 | Primitive of x over 1 minus Sine of a x | :$\ds \int \frac {x \rd x} {1 - \sin a x} = \frac x a \map \tan {\frac \pi 4 + \frac {a x} 2} + \frac 2 {a^2} \ln \size {\map \sin {\frac \pi 4 - \frac {a x} 2} } + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int \frac {x \rd x} {1 - \sin a x} = \frac x a \map \tan {\frac \pi 4 + \frac {a x} 2} + \frac 2 {a^2} \ln \size {\map \sin {\frac \pi 4 - \frac {a x} 2} } + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x over 1 minus Sine of a x | https://proofwiki.org/wiki/Primitive_of_x_over_1_minus_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_over_1_minus_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Reciprocal of 1 minus Sine of a x",
"Integration by Parts",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Tangent Function/Cosine Form",
"Sine of Complement equals Cosine"
] |
proofwiki-9516 | Primitive of Reciprocal of 1 plus Sine of a x | :$\ds \int \frac {\d x} {1 + \sin a x} = -\frac 1 a \map \tan {\frac \pi 4 - \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {1 + \sin a x}
| r = \int \frac 1 2 \map {\sec^2} {\frac \pi 4 - \frac {a x} 2} \rd x
| c = Reciprocal of One Plus Sine
}}
{{eqn | n = 1
| r = \frac 1 2 \int \map {\sec^2} {\frac \pi 4 - \frac {a x} 2} \rd x
| c = Primitive of Constant Multiple of Func... | :$\ds \int \frac {\d x} {1 + \sin a x} = -\frac 1 a \map \tan {\frac \pi 4 - \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {1 + \sin a x}
| r = \int \frac 1 2 \map {\sec^2} {\frac \pi 4 - \frac {a x} 2} \rd x
| c = [[Reciprocal of One Plus Sine]]
}}
{{eqn | n = 1
| r = \frac 1 2 \int \map {\sec^2} {\frac \pi 4 - \frac {a x} 2} \rd x
| c = [[Primitive of Constant Multiple o... | Primitive of Reciprocal of 1 plus Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_plus_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_plus_Sine_of_a_x | [
"Primitive of Reciprocal of 1 plus Sine of a x",
"Primitives involving Sine Function"
] | [] | [
"Reciprocal of One Plus Sine",
"Primitive of Constant Multiple of Function",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Square of Secant Function"
] |
proofwiki-9517 | Primitive of x over 1 plus Sine of a x | :$\ds \int \frac {x \rd x} {1 + \sin a x} = -\frac x a \map \tan {\frac \pi 4 - \frac {a x} 2} + \frac 2 {a^2} \ln \size {\map \sin {\frac \pi 4 + \frac {a x} 2} } + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int \frac {x \rd x} {1 + \sin a x} = -\frac x a \map \tan {\frac \pi 4 - \frac {a x} 2} + \frac 2 {a^2} \ln \size {\map \sin {\frac \pi 4 + \frac {a x} 2} } + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x over 1 plus Sine of a x | https://proofwiki.org/wiki/Primitive_of_x_over_1_plus_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_over_1_plus_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Reciprocal of 1 plus Sine of a x",
"Integration by Parts",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Tangent Function/Cosine Form",
"Sine of Complement equals Cosine"
] |
proofwiki-9518 | Primitive of Reciprocal of Square of 1 minus Sine of a x | :$\ds \int \frac {\d x} {\paren {1 - \sin a x}^2} = \frac 1 {2 a} \map \tan {\frac \pi 4 + \frac {a x} 2} + \frac 1 {6 a} \map {\tan^3} {\frac \pi 4 + \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {1 - \sin a x}^2}
| r = \int \paren {\frac 1 2 \map {\sec^2} {\frac \pi 4 + \frac {a x} 2} }^2 \rd x
| c = Reciprocal of One Minus Sine
}}
{{eqn | r = \frac 1 4 \int \map {\sec^4} {\frac \pi 4 + \frac {a x} 2} \rd x
| c = simplifying
}}
{{end-eqn}}
L... | :$\ds \int \frac {\d x} {\paren {1 - \sin a x}^2} = \frac 1 {2 a} \map \tan {\frac \pi 4 + \frac {a x} 2} + \frac 1 {6 a} \map {\tan^3} {\frac \pi 4 + \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {1 - \sin a x}^2}
| r = \int \paren {\frac 1 2 \map {\sec^2} {\frac \pi 4 + \frac {a x} 2} }^2 \rd x
| c = [[Reciprocal of One Minus Sine]]
}}
{{eqn | r = \frac 1 4 \int \map {\sec^4} {\frac \pi 4 + \frac {a x} 2} \rd x
| c = simplifying
}}
{{end-eqn... | Primitive of Reciprocal of Square of 1 minus Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_1_minus_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_1_minus_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Reciprocal of One Minus Sine",
"Integration by Substitution",
"Primitive of Power of Secant of a x",
"Primitive of Square of Secant of a x",
"Sum of Squares of Sine and Cosine/Corollary 1"
] |
proofwiki-9519 | Reciprocal of One Minus Sine | :$\dfrac 1 {1 - \sin x} = \dfrac 1 2 \map {\sec^2} {\dfrac \pi 4 + \dfrac x 2}$ | {{begin-eqn}}
{{eqn | l = 1 - \sin x
| r = \sin \frac \pi 2 - \sin x
| c = Sine of Right Angle
}}
{{eqn | r = 2 \map \cos {\frac 1 2 \paren {\frac \pi 2 + x} } \map \sin {\frac 1 2 \paren {\frac \pi 2 - x} }
| c = Sine minus Sine
}}
{{eqn | r = 2 \map \cos {\frac \pi 4 + \frac x 2} \map \sin {\frac \p... | :$\dfrac 1 {1 - \sin x} = \dfrac 1 2 \map {\sec^2} {\dfrac \pi 4 + \dfrac x 2}$ | {{begin-eqn}}
{{eqn | l = 1 - \sin x
| r = \sin \frac \pi 2 - \sin x
| c = [[Sine of Right Angle]]
}}
{{eqn | r = 2 \map \cos {\frac 1 2 \paren {\frac \pi 2 + x} } \map \sin {\frac 1 2 \paren {\frac \pi 2 - x} }
| c = [[Sine minus Sine]]
}}
{{eqn | r = 2 \map \cos {\frac \pi 4 + \frac x 2} \map \sin {... | Reciprocal of One Minus Sine | https://proofwiki.org/wiki/Reciprocal_of_One_Minus_Sine | https://proofwiki.org/wiki/Reciprocal_of_One_Minus_Sine | [
"Trigonometric Identities",
"Sine Function"
] | [] | [
"Sine of Right Angle",
"Prosthaphaeresis Formulas/Sine minus Sine",
"Cosine of Complement equals Sine"
] |
proofwiki-9520 | Reciprocal of One Plus Sine | :$\dfrac 1 {1 + \sin x} = \dfrac 1 2 \map {\sec^2} {\dfrac \pi 4 - \dfrac x 2}$ | {{begin-eqn}}
{{eqn | l = 1 + \sin x
| r = \sin \frac \pi 2 + \sin x
| c = Sine of Right Angle
}}
{{eqn | r = 2 \map \sin {\frac 1 2 \paren {\frac \pi 2 + x} } \map \cos {\frac 1 2 \paren {\frac \pi 2 - x} }
| c = Sine minus Sine
}}
{{eqn | r = 2 \map \sin {\frac \pi 4 + \frac x 2} \map \cos {\frac \p... | :$\dfrac 1 {1 + \sin x} = \dfrac 1 2 \map {\sec^2} {\dfrac \pi 4 - \dfrac x 2}$ | {{begin-eqn}}
{{eqn | l = 1 + \sin x
| r = \sin \frac \pi 2 + \sin x
| c = [[Sine of Right Angle]]
}}
{{eqn | r = 2 \map \sin {\frac 1 2 \paren {\frac \pi 2 + x} } \map \cos {\frac 1 2 \paren {\frac \pi 2 - x} }
| c = [[Sine minus Sine]]
}}
{{eqn | r = 2 \map \sin {\frac \pi 4 + \frac x 2} \map \cos {... | Reciprocal of One Plus Sine | https://proofwiki.org/wiki/Reciprocal_of_One_Plus_Sine | https://proofwiki.org/wiki/Reciprocal_of_One_Plus_Sine | [
"Trigonometric Identities",
"Sine Function"
] | [] | [
"Sine of Right Angle",
"Prosthaphaeresis Formulas/Sine minus Sine",
"Cosine of Complement equals Sine"
] |
proofwiki-9521 | Power Series Expansion for Cosecant Function | The cosecant function has a Laurent series expansion:
{{begin-eqn}}
{{eqn | l = \csc x
| r = \sum_{n \mathop = 0}^\infty \dfrac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 x + \frac x 6 + \frac {7 x^3} {360} + \frac {31 x^5} {15 \, 1... | {{begin-eqn}}
{{eqn | l = \sin x
| r = 2 \sin \dfrac x 2 \cos \dfrac x 2
| c = Double Angle Formula for Sine
}}
{{eqn | ll= \leadstoandfrom
| l = \dfrac 1 {\sin x}
| r = \dfrac 1 {2 \sin \dfrac x 2 \cos \dfrac x 2}
| c = taking the reciprocal of both sides
}}
{{eqn | ll= \leadstoandfrom
... | The [[Definition:Cosecant|cosecant function]] has a [[Definition:Laurent Series|Laurent series expansion]]:
{{begin-eqn}}
{{eqn | l = \csc x
| r = \sum_{n \mathop = 0}^\infty \dfrac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 x + \f... | {{begin-eqn}}
{{eqn | l = \sin x
| r = 2 \sin \dfrac x 2 \cos \dfrac x 2
| c = [[Double Angle Formula for Sine]]
}}
{{eqn | ll= \leadstoandfrom
| l = \dfrac 1 {\sin x}
| r = \dfrac 1 {2 \sin \dfrac x 2 \cos \dfrac x 2}
| c = taking the reciprocal of both sides
}}
{{eqn | ll= \leadstoandfro... | Power Series Expansion for Cosecant Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Cosecant_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Cosecant_Function | [
"Power Series Expansion for Cosecant Function",
"Examples of Power Series",
"Bernoulli Numbers",
"Cosecant Function"
] | [
"Definition:Cosecant",
"Definition:Laurent Series",
"Definition:Bernoulli Numbers",
"Definition:Convergent Series"
] | [
"Double Angle Formulas/Sine",
"Cosecant is Reciprocal of Sine",
"Secant is Reciprocal of Cosine",
"Euler's Secant Identity",
"Euler's Cosecant Identity",
"Odd Bernoulli Numbers Vanish"
] |
proofwiki-9522 | Periodic Function plus Constant | Let $f: \R \to \R$ be a real function.
Let $k \in \R$ be constant.
Then $f$ is periodic with period $L$ {{iff}} $f + k$ is periodic with period $L$. | === Sufficient Condition ===
Let $f$ be periodic with period $L$.
Then:
{{begin-eqn}}
{{eqn | l = \map f x
| r = \map f {x + L}
| c = {{Defof|Periodic Real Function}}
}}
{{eqn | ll= \leadsto
| l = \map f x + k
| r = \map f {x + L} + k
}}
{{end-eqn}}
Thus $f + k$ has been shown to be periodic wit... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
Let $k \in \R$ be [[Definition:Constant|constant]].
Then $f$ is [[Definition:Periodic Real Function|periodic]] with [[Definition:Period of Periodic Real Function|period]] $L$ {{iff}} $f + k$ is [[Definition:Periodic Real Function|periodic]] with [[D... | === Sufficient Condition ===
Let $f$ be [[Definition:Periodic Real Function|periodic]] with [[Definition:Period of Periodic Real Function|period]] $L$.
Then:
{{begin-eqn}}
{{eqn | l = \map f x
| r = \map f {x + L}
| c = {{Defof|Periodic Real Function}}
}}
{{eqn | ll= \leadsto
| l = \map f x + k
... | Periodic Function plus Constant | https://proofwiki.org/wiki/Periodic_Function_plus_Constant | https://proofwiki.org/wiki/Periodic_Function_plus_Constant | [
"Periodic Functions"
] | [
"Definition:Real Function",
"Definition:Constant",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period"
] | [
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Periodic Function/Real",
"Definition:Periodic R... |
proofwiki-9523 | Derivative of Periodic Real Function | Let $f: \R \to \R$ be a real function.
Let $f$ be differentiable on all of $\R$.
Let $f$ be periodic with period $L$.
Then its derivative is also periodic with period $L$. | Let $f$ be differentiable on all of $\R$.
Let $f$ be periodic with period $L$.
Then taking the derivative of both sides using the Chain Rule for Derivatives yields:
:$\map f x = \map f {x + L} \implies \map {f'} x = \map {f'} {x + L}$
Let $L'$ be the period of $f'$.
$f$ is differentiable and therefore continuous, by Di... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
Let $f$ be [[Definition:Differentiable Real Function|differentiable]] on all of $\R$.
Let $f$ be [[Definition:Periodic Real Function|periodic]] with [[Definition:Period of Periodic Real Function|period]] $L$.
Then its [[Definition:Derivative|deriva... | Let $f$ be [[Definition:Differentiable Real Function|differentiable]] on all of $\R$.
Let $f$ be [[Definition:Periodic Real Function|periodic]] with [[Definition:Period of Periodic Real Function|period]] $L$.
Then taking the [[Definition:Derivative|derivative]] of both sides using the [[Chain Rule for Derivatives]] ... | Derivative of Periodic Real Function | https://proofwiki.org/wiki/Derivative_of_Periodic_Real_Function | https://proofwiki.org/wiki/Derivative_of_Periodic_Real_Function | [
"Differential Calculus",
"Periodic Functions"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Derivative",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period"
] | [
"Definition:Differentiable Mapping/Real Function",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Derivative",
"Derivative of Composite Function",
"Definition:Periodic Real Function/Period",
"Definition:Differentiable Mapping/Real Function",
"Definition:Co... |
proofwiki-9524 | Sum of Two Odd Powers/Examples/Sum of Two Cubes | :$x^3 + y^3 = \paren {x + y} \paren {x^2 - x y + y^2}$ | From Difference of Two Powers:
:$\ds a^n - b^n = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j$
Let $x = a$ and $y = -b$.
Then:
{{begin-eqn}}
{{eqn | l = x^3 + y^3
| r = x^3 - \paren {-y^3}
| c =
}}
{{eqn | r = x^3 - \paren {-y}^3
| c =
}}
{{eqn | r = \paren {x - \paren {-y} } \sum_{... | :$x^3 + y^3 = \paren {x + y} \paren {x^2 - x y + y^2}$ | From [[Difference of Two Powers]]:
:$\ds a^n - b^n = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j$
Let $x = a$ and $y = -b$.
Then:
{{begin-eqn}}
{{eqn | l = x^3 + y^3
| r = x^3 - \paren {-y^3}
| c =
}}
{{eqn | r = x^3 - \paren {-y}^3
| c =
}}
{{eqn | r = \paren {x - \paren {-y} } ... | Sum of Two Odd Powers/Examples/Sum of Two Cubes/Proof 1 | https://proofwiki.org/wiki/Sum_of_Two_Odd_Powers/Examples/Sum_of_Two_Cubes | https://proofwiki.org/wiki/Sum_of_Two_Odd_Powers/Examples/Sum_of_Two_Cubes/Proof_1 | [
"Sum of Two Cubes",
"Examples of Use of Sum of Two Odd Powers"
] | [] | [
"Difference of Two Powers"
] |
proofwiki-9525 | Sum of Two Odd Powers/Examples/Sum of Two Cubes | :$x^3 + y^3 = \paren {x + y} \paren {x^2 - x y + y^2}$ | From Sum of Two Odd Powers:
:$a^{2 n + 1} + b^{2 n + 1} = \paren {a + b} \paren {a^{2 n} - a^{2 n - 1} b + a^{2 n - 2} b^2 - \dotsb + a b^{2 n - 1} + b^{2 n} }$
We have that $3 = 2 \times 1 + 1$.
Hence setting $n = 1$ gives the result.
{{qed}} | :$x^3 + y^3 = \paren {x + y} \paren {x^2 - x y + y^2}$ | From [[Sum of Two Odd Powers]]:
:$a^{2 n + 1} + b^{2 n + 1} = \paren {a + b} \paren {a^{2 n} - a^{2 n - 1} b + a^{2 n - 2} b^2 - \dotsb + a b^{2 n - 1} + b^{2 n} }$
We have that $3 = 2 \times 1 + 1$.
Hence setting $n = 1$ gives the result.
{{qed}} | Sum of Two Odd Powers/Examples/Sum of Two Cubes/Proof 2 | https://proofwiki.org/wiki/Sum_of_Two_Odd_Powers/Examples/Sum_of_Two_Cubes | https://proofwiki.org/wiki/Sum_of_Two_Odd_Powers/Examples/Sum_of_Two_Cubes/Proof_2 | [
"Sum of Two Cubes",
"Examples of Use of Sum of Two Odd Powers"
] | [] | [
"Sum of Two Odd Powers"
] |
proofwiki-9526 | Primitive of Periodic Real Function | Let $f: \R \to \R$ be a real function.
Let $F$ be a primitive of $f$ that is bounded on all of $\R$.
Let $f$ be periodic with period $L$.
Then $F$ is also periodic with period $L$. | {{MissingLinks}}
Let $f$ be periodic with period $L$.
Let $f$ have a primitive $F$ that is bounded on all of $\R$.
By definition of a periodic function, it is seen that:
:$\map f x = \map f {x + L}$.
Then:
:$\ds \int \map f x \rd x$
and:
:$\ds \int \map f {x + L} \rd x$
are both primitives of the same function.
So by P... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
Let $F$ be a [[Definition:Primitive (Calculus)|primitive]] of $f$ that is [[Definition:Bounded Mapping|bounded]] on all of $\R$.
Let $f$ be [[Definition:Real Periodic Function|periodic]] with [[Definition:Period of Periodic Real Function|period]] $L$... | {{MissingLinks}}
Let $f$ be [[Definition:Real Periodic Function|periodic]] with [[Definition:Period of Periodic Real Function|period]] $L$.
Let $f$ have a [[Definition:Primitive (Calculus)|primitive]] $F$ that is [[Definition:Bounded Mapping|bounded]] on all of $\R$.
By definition of a [[Definition:Real Periodic Fun... | Primitive of Periodic Real Function | https://proofwiki.org/wiki/Primitive_of_Periodic_Real_Function | https://proofwiki.org/wiki/Primitive_of_Periodic_Real_Function | [
"Primitives",
"Periodic Functions"
] | [
"Definition:Real Function",
"Definition:Primitive (Calculus)",
"Definition:Bounded Mapping",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period"
] | [
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Primitive (Calculus)",
"Definition:Bounded Mapping",
"Definition:Periodic Function/Real",
"Definition:Primitive (Calculus)",
"Primitives which Differ by Constant",
"Integration by Substitution",
"Principle o... |
proofwiki-9527 | Primitive of Reciprocal of x squared by x cubed plus a cubed/Lemma | :$\ds \int \frac {\d x} {x^2 \paren {x^3 + a^3} } = \frac {-1} {a^3 x} - \frac 1 {a^3} \int \frac {x \rd x} {\paren {x^3 + a^3} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {x^3 + a^3} }
| r = \int \frac {a^3 \rd x} {a^3 x^2 \paren {x^3 + a^3} }
| c = multiplying top and bottom by $a^3$
}}
{{eqn | r = \int \frac {\paren {x^3 + a^3 - x^3} \rd x} {a^3 x^2 \paren {x^3 + a^3} }
| c =
}}
{{eqn | r = \frac 1 {a^3} \int \... | :$\ds \int \frac {\d x} {x^2 \paren {x^3 + a^3} } = \frac {-1} {a^3 x} - \frac 1 {a^3} \int \frac {x \rd x} {\paren {x^3 + a^3} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {x^3 + a^3} }
| r = \int \frac {a^3 \rd x} {a^3 x^2 \paren {x^3 + a^3} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a^3$
}}
{{eqn | r = \int \frac {\paren {x^3 + a^3 - x^3} \rd x} {a^3 x^2 \paren {x^3 + a^3... | Primitive of Reciprocal of x squared by x cubed plus a cubed/Lemma | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_cubed_plus_a_cubed/Lemma | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_cubed_plus_a_cubed/Lemma | [
"Primitives involving x cubed plus a cubed/Lemmata",
"Primitive of Reciprocal of x squared by x cubed plus a cubed"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Category:Primitives involving x cubed plus a cubed/Lemmata",
"Category:Primitive of Reciprocal of x squared by x cubed plus a cubed"
] |
proofwiki-9528 | Sum of Two Fourth Powers | :$x^4 + y^4 = \paren {x^2 + \sqrt 2 x y + y^2} \paren {x^2 - \sqrt 2 x y + y^2}$ | {{begin-eqn}}
{{eqn | r = \paren {x^2 + \sqrt 2 x y + y^2} \paren {x^2 - \sqrt 2 x y + y^2}
| o =
| c =
}}
{{eqn | r = x^2 \paren {x^2 - \sqrt 2 x y + y^2} + \sqrt 2 x y \paren {x^2 - \sqrt 2 x y + y^2} + y^2 \paren {x^2 - \sqrt 2 x y + y^2}
| c =
}}
{{eqn | r = x^4 - \sqrt 2 x^3 y + x^2 y^2 + \sqr... | :$x^4 + y^4 = \paren {x^2 + \sqrt 2 x y + y^2} \paren {x^2 - \sqrt 2 x y + y^2}$ | {{begin-eqn}}
{{eqn | r = \paren {x^2 + \sqrt 2 x y + y^2} \paren {x^2 - \sqrt 2 x y + y^2}
| o =
| c =
}}
{{eqn | r = x^2 \paren {x^2 - \sqrt 2 x y + y^2} + \sqrt 2 x y \paren {x^2 - \sqrt 2 x y + y^2} + y^2 \paren {x^2 - \sqrt 2 x y + y^2}
| c =
}}
{{eqn | r = x^4 - \sqrt 2 x^3 y + x^2 y^2 + \sqr... | Sum of Two Fourth Powers | https://proofwiki.org/wiki/Sum_of_Two_Fourth_Powers | https://proofwiki.org/wiki/Sum_of_Two_Fourth_Powers | [
"Fourth Powers"
] | [] | [
"Category:Fourth Powers"
] |
proofwiki-9529 | Pumping Lemma for Regular Languages | Let $\LL_3$ be the set of regular languages.
{{explain|Is it a set? Does this need to be proved? Intuition would suggest that it would be a class.}}
Then the following holds:
$\forall L \in \LL_3: \exists n_0 \in \N_0: \forall z \in L: \card z > n_0 \implies \exists u, v, w$ such that:
:$z = u \cdot v \cdot w$
:$\card ... | === For finite languages ===
For any ''finite'' regular language $L_{fin}$, the proof is simple.
Let $s_{maxlen} \in L_{fin}$.
Thus:
:$\forall s \in L_{fin}: \card s \le \card {s_{maxlen} }$
Now choose $n_0 > \card {s_{maxlen} }$.
The implication now trivially holds because the premise:
:$\paren {\card z > n_0}$
is fal... | Let $\LL_3$ be the set of regular languages.
{{explain|Is it a set? Does this need to be proved? Intuition would suggest that it would be a class.}}
Then the following holds:
$\forall L \in \LL_3: \exists n_0 \in \N_0: \forall z \in L: \card z > n_0 \implies \exists u, v, w$ such that:
:$z = u \cdot v \cdot w$
:$\c... | === For finite languages ===
For any ''finite'' regular language $L_{fin}$, the proof is simple.
Let $s_{maxlen} \in L_{fin}$.
Thus:
:$\forall s \in L_{fin}: \card s \le \card {s_{maxlen} }$
Now choose $n_0 > \card {s_{maxlen} }$.
The implication now trivially holds because the premise:
:$\paren {\card z > n_0}$
i... | Pumping Lemma for Regular Languages | https://proofwiki.org/wiki/Pumping_Lemma_for_Regular_Languages | https://proofwiki.org/wiki/Pumping_Lemma_for_Regular_Languages | [
"Computer Science",
"Named Theorems"
] | [] | [] |
proofwiki-9530 | One Plus Tangent Half Angle over One Minus Tangent Half Angle | :$\dfrac {1 + \tan \frac x 2} {1 - \tan \frac x 2} = \sec x + \tan x$ | {{begin-eqn}}
{{eqn | l = \frac {1 + \tan \frac x 2}{1 - \tan \frac x 2}
| r = \frac {1 + \frac {\sin \frac x 2}{\cos \frac x 2} }{1 - \frac {\sin \frac x 2}{\cos \frac x 2} }
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {\cos \frac x 2 + \sin \frac x 2}{\cos \frac x 2 - \sin \frac x 2}
... | :$\dfrac {1 + \tan \frac x 2} {1 - \tan \frac x 2} = \sec x + \tan x$ | {{begin-eqn}}
{{eqn | l = \frac {1 + \tan \frac x 2}{1 - \tan \frac x 2}
| r = \frac {1 + \frac {\sin \frac x 2}{\cos \frac x 2} }{1 - \frac {\sin \frac x 2}{\cos \frac x 2} }
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {\cos \frac x 2 + \sin \frac x 2}{\cos \frac x 2 - \sin \frac x 2}
... | One Plus Tangent Half Angle over One Minus Tangent Half Angle | https://proofwiki.org/wiki/One_Plus_Tangent_Half_Angle_over_One_Minus_Tangent_Half_Angle | https://proofwiki.org/wiki/One_Plus_Tangent_Half_Angle_over_One_Minus_Tangent_Half_Angle | [
"Trigonometric Identities"
] | [] | [
"Tangent is Sine divided by Cosine",
"Sum of Squares of Sine and Cosine",
"Double Angle Formulas/Sine",
"Double Angle Formulas/Cosine",
"Sum of Secant and Tangent",
"Category:Trigonometric Identities"
] |
proofwiki-9531 | Primitive of Power of Cosecant of a x | :$\ds \int \csc^n a x \rd x = \frac{-\csc^{n - 2} a x \cot a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \csc^{n - 2} a x \rd x$
where $n \ne -1$. | 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 = \csc^{n - 2} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -a \paren {n - 2} \csc^{n - 3} a x \csc a x... | :$\ds \int \csc^n a x \rd x = \frac{-\csc^{n - 2} a x \cot a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \csc^{n - 2} a x \rd x$
where $n \ne -1$. | 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 = \csc^{n - 2} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -a \paren {n - 2}... | Primitive of Power of Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Cosecant_of_a_x | [
"Primitives involving Cosecant Function"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Derivative of Cosecant Function",
"Derivative of Composite Function",
"Primitive of Square of Cosecant of a x",
"Integration by Parts",
"Sum of Squares of Sine and Cosine/Corollary 2",
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9532 | Primitive of Cosecant of a x/Tangent Form | :$\ds \int \csc a x \rd x = \frac 1 a \ln \size {\tan \frac {a x} 2} + C$
where $\tan \dfrac {a x} 2 \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \csc x \rd x
| r = \ln \size {\tan \frac x 2}
| c = Primitive of $\csc x$: Tangent Form
}}
{{eqn | ll= \leadsto
| l = \int \csc a x \rd x
| r = \frac 1 a \ln \size {\tan \frac {a x} 2} + C
| c = Primitive of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}... | :$\ds \int \csc a x \rd x = \frac 1 a \ln \size {\tan \frac {a x} 2} + C$
where $\tan \dfrac {a x} 2 \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \csc x \rd x
| r = \ln \size {\tan \frac x 2}
| c = [[Primitive of Cosecant Function/Tangent Form|Primitive of $\csc x$: Tangent Form]]
}}
{{eqn | ll= \leadsto
| l = \int \csc a x \rd x
| r = \frac 1 a \ln \size {\tan \frac {a x} 2} + C
| c = [[Primitive of F... | Primitive of Cosecant of a x/Tangent Form | https://proofwiki.org/wiki/Primitive_of_Cosecant_of_a_x/Tangent_Form | https://proofwiki.org/wiki/Primitive_of_Cosecant_of_a_x/Tangent_Form | [
"Primitive of Cosecant of a x"
] | [] | [
"Primitive of Cosecant Function/Tangent Form",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9533 | Primitive of Cosecant of a x/Cosecant minus Cotangent Form | :$\ds \int \csc a x \rd x = \frac 1 a \ln \size {\csc a x - \cot a x} + C$
where $\csc a x - \cot a x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \csc x \rd x
| r = \ln \size {\csc x - \cot x}
| c = Primitive of $\csc x$: Cosecant minus Cotangent Form
}}
{{eqn | ll= \leadsto
| l = \int \csc a x \rd x
| r = \frac 1 a \ln \size {\csc a x - \cot a x} + C
| c = Primitive of Function of Constant Multiple
}}... | :$\ds \int \csc a x \rd x = \frac 1 a \ln \size {\csc a x - \cot a x} + C$
where $\csc a x - \cot a x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \csc x \rd x
| r = \ln \size {\csc x - \cot x}
| c = [[Primitive of Cosecant Function/Cosecant minus Cotangent Form|Primitive of $\csc x$: Cosecant minus Cotangent Form]]
}}
{{eqn | ll= \leadsto
| l = \int \csc a x \rd x
| r = \frac 1 a \ln \size {\csc a x - \cot a... | Primitive of Cosecant of a x/Cosecant minus Cotangent Form | https://proofwiki.org/wiki/Primitive_of_Cosecant_of_a_x/Cosecant_minus_Cotangent_Form | https://proofwiki.org/wiki/Primitive_of_Cosecant_of_a_x/Cosecant_minus_Cotangent_Form | [
"Primitive of Cosecant of a x"
] | [] | [
"Primitive of Cosecant Function/Cosecant minus Cotangent Form",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9534 | Primitive of Reciprocal of Square of 1 plus Sine of a x | :$\ds \int \frac {\d x} {\paren {1 + \sin a x}^2} = \frac {-1} {2 a} \map \tan {\frac \pi 4 - \frac {a x} 2} - \frac 1 {6 a} \map {\tan^3} {\frac \pi 4 - \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {1 + \sin a x}^2}
| r = \int \paren {\frac 1 2 \map {\sec^2} {\dfrac \pi 4 - \frac {a x} 2} }^2 \rd x
| c = Reciprocal of One Plus Sine
}}
{{eqn | r = \frac 1 4 \int \map {\sec^4} {\frac \pi 4 - \frac {a x} 2} \rd x
| c = simplifying
}}
{{end-eqn}}
L... | :$\ds \int \frac {\d x} {\paren {1 + \sin a x}^2} = \frac {-1} {2 a} \map \tan {\frac \pi 4 - \frac {a x} 2} - \frac 1 {6 a} \map {\tan^3} {\frac \pi 4 - \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {1 + \sin a x}^2}
| r = \int \paren {\frac 1 2 \map {\sec^2} {\dfrac \pi 4 - \frac {a x} 2} }^2 \rd x
| c = [[Reciprocal of One Plus Sine]]
}}
{{eqn | r = \frac 1 4 \int \map {\sec^4} {\frac \pi 4 - \frac {a x} 2} \rd x
| c = simplifying
}}
{{end-eqn... | Primitive of Reciprocal of Square of 1 plus Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_1_plus_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_1_plus_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Reciprocal of One Plus Sine",
"Integration by Substitution",
"Primitive of Power of Secant of a x",
"Primitive of Square of Secant of a x",
"Sum of Squares of Sine and Cosine/Corollary 1"
] |
proofwiki-9535 | Primitive of Reciprocal of p plus q by Sine of a x | :$\ds \int \frac {\d x} {p + q \sin a x} = \begin{cases}
\ds \frac 2 {a \sqrt {p^2 - q^2} } \map \arctan {\frac {p \tan \dfrac {a x} 2 + q} {\sqrt {p^2 - q^2} } } + C & : q^2 - p^2 < 0 \\
\ds \frac 1 {a \sqrt {q^2 - p^2} } \ln \size {\frac {p \tan \dfrac {a x} 2 + q - \sqrt {p^2 - q^2} } {p \tan \dfrac {a x} 2 + q + \... | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \sin a x}
| r = \frac 2 a \int \frac {\d u} {p u^2 + 2 q u + p}
| c = Weierstrass Substitution: $u = \tan \dfrac {a x} 2$
}}
{{end-eqn}}
The discriminant of $p u^2 + 2 q u + p$ is $4 q^2 - 4 p^2$.
Thus:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \s... | :$\ds \int \frac {\d x} {p + q \sin a x} = \begin{cases}
\ds \frac 2 {a \sqrt {p^2 - q^2} } \map \arctan {\frac {p \tan \dfrac {a x} 2 + q} {\sqrt {p^2 - q^2} } } + C & : q^2 - p^2 < 0 \\
\ds \frac 1 {a \sqrt {q^2 - p^2} } \ln \size {\frac {p \tan \dfrac {a x} 2 + q - \sqrt {p^2 - q^2} } {p \tan \dfrac {a x} 2 + q + \... | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \sin a x}
| r = \frac 2 a \int \frac {\d u} {p u^2 + 2 q u + p}
| c = [[Primitive of Reciprocal of p plus q by Sine of a x/Weierstrass Substitution|Weierstrass Substitution]]: $u = \tan \dfrac {a x} 2$
}}
{{end-eqn}}
The [[Definition:Discriminant of Poly... | Primitive of Reciprocal of p plus q by Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Sine_of_a_x | [
"Primitive of Reciprocal of p plus q by Sine of a x",
"Primitives involving Sine Function"
] | [] | [
"Primitive of Reciprocal of p plus q by Sine of a x/Weierstrass Substitution",
"Definition:Discriminant of Polynomial",
"Primitive of Reciprocal of a x squared plus b x plus c"
] |
proofwiki-9536 | Primitive of Reciprocal of square of p plus q by Sine of a x | :$\ds \int \frac {\d x} {\paren {p + q \sin a x}^2} = \frac {q \cos a x} {a \paren {p^2 - q^2} \paren {p + q \sin a x} } + \frac p {p^2 - q^2} \int \frac {\d x} {p + q \sin a x}$ | First a pair of lemmata:
=== Lemma ===
{{:Primitive of Reciprocal of square of p plus q by Sine of a x/Lemma}}{{qed|lemma}}
=== Weierstrass Substitution ===
{{:Primitive of Reciprocal of square of p plus q by Sine of a x/Weierstrass Substitution|Weierstrass Substitution}}{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = \int \fr... | :$\ds \int \frac {\d x} {\paren {p + q \sin a x}^2} = \frac {q \cos a x} {a \paren {p^2 - q^2} \paren {p + q \sin a x} } + \frac p {p^2 - q^2} \int \frac {\d x} {p + q \sin a x}$ | First a pair of [[Definition:Lemma|lemmata]]:
=== [[Primitive of Reciprocal of square of p plus q by Sine of a x/Lemma|Lemma]] ===
{{:Primitive of Reciprocal of square of p plus q by Sine of a x/Lemma}}{{qed|lemma}}
=== [[Primitive of Reciprocal of square of p plus q by Sine of a x/Weierstrass Substitution|Weierstra... | Primitive of Reciprocal of square of p plus q by Sine of a x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_square_of_p_plus_q_by_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_square_of_p_plus_q_by_Sine_of_a_x/Proof_2 | [
"Primitives involving Sine Function",
"Primitive of Reciprocal of square of p plus q by Sine of a x"
] | [] | [
"Definition:Lemma",
"Primitive of Reciprocal of square of p plus q by Sine of a x/Lemma",
"Primitive of Reciprocal of square of p plus q by Sine of a x/Weierstrass Substitution",
"Primitive of Reciprocal of square of p plus q by Sine of a x/Weierstrass Substitution",
"Linear Combination of Integrals/Indefin... |
proofwiki-9537 | Primitive of Reciprocal of p squared plus square of q by Sine of a x | :$\ds \int \frac {\d x} {p^2 + q^2 \sin^2 a x} = \frac 1 {a p \sqrt {p^2 + q^2} } \arctan \frac {\sqrt {p^2 + q^2} \tan a x} p + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 + q^2 \sin^2 a x}
| r = \int \frac {\sec^2 a x \rd x} {p^2 \sec^2 a x + q^2 \tan^2 a x}
| c = multiplying numerator and denominator by $\sec^2 a x$
}}
{{eqn | r = \int \frac {\sec^2 a x \rd x} {p^2 + \paren {p^2 + q^2} \tan^2 a x}
| c = Difference of Sq... | :$\ds \int \frac {\d x} {p^2 + q^2 \sin^2 a x} = \frac 1 {a p \sqrt {p^2 + q^2} } \arctan \frac {\sqrt {p^2 + q^2} \tan a x} p + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 + q^2 \sin^2 a x}
| r = \int \frac {\sec^2 a x \rd x} {p^2 \sec^2 a x + q^2 \tan^2 a x}
| c = multiplying [[Definition:Numerator|numerator]] and [[Definition:Denominator|denominator]] by $\sec^2 a x$
}}
{{eqn | r = \int \frac {\sec^2 a x \rd x} {p^2 + \paren ... | Primitive of Reciprocal of p squared plus square of q by Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_plus_square_of_q_by_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_plus_square_of_q_by_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Derivative of Tangent Function",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-9538 | Primitive of Reciprocal of p squared minus square of q by Sine of a x | $\quad \ds \int \frac {\d x} {p^2 - q^2 \sin^2 a x} = \begin {cases}
\dfrac 1 {a p \sqrt {p^2 - q^2} } \arctan \dfrac {\sqrt {p^2 - q^2} \tan a x} p & : p^2 > q^2 \\
\dfrac 1 {2 a p \sqrt {q^2 - p^2} } \ln \size {\dfrac {\sqrt {q^2 - p^2} \tan a x + p} {\sqrt {q^2 - p^2} \tan a x - p} } & : p^2 < q^2
\end {cases}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 - q^2 \sin^2 a x}
| r = \int \frac {\sec^2 a x \rd x} {p^2 \sec^2 a x - q^2 \tan^2 a x}
| c = multiplying numerator and denominator by $\sec^2 a x$
}}
{{eqn | r = \int \frac {\sec^2 a x \rd x} {p^2 \paren {1 + \tan^2 a x} - q^2 \tan^2 a x}
| c = Differe... | $\quad \ds \int \frac {\d x} {p^2 - q^2 \sin^2 a x} = \begin {cases}
\dfrac 1 {a p \sqrt {p^2 - q^2} } \arctan \dfrac {\sqrt {p^2 - q^2} \tan a x} p & : p^2 > q^2 \\
\dfrac 1 {2 a p \sqrt {q^2 - p^2} } \ln \size {\dfrac {\sqrt {q^2 - p^2} \tan a x + p} {\sqrt {q^2 - p^2} \tan a x - p} } & : p^2 < q^2
\end {cases}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 - q^2 \sin^2 a x}
| r = \int \frac {\sec^2 a x \rd x} {p^2 \sec^2 a x - q^2 \tan^2 a x}
| c = multiplying [[Definition:Numerator|numerator]] and [[Definition:Denominator|denominator]] by $\sec^2 a x$
}}
{{eqn | r = \int \frac {\sec^2 a x \rd x} {p^2 \paren {1... | Primitive of Reciprocal of p squared minus square of q by Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_minus_square_of_q_by_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_minus_square_of_q_by_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Derivative of Tangent Function",
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Primitive of Reciprocal of x squared minus a squar... |
proofwiki-9539 | Primitive of Power of x by Sine of a x | :$\ds \int x^m \sin a x \rd x = \frac {-x^m \cos a x} a + \frac {m x^{m - 1} \sin a x} {a^2} - \frac {m \paren {m - 1} } {a^2} \int x^{m - 2} \sin a x \rd x$ | === Lemma ===
{{:Primitive of Power of x by Sine of a x/Lemma}}{{qed|lemma}}
From {{Lemma|Primitive of Power of x by Cosine of a x|proof = yes|disp = Primitive of $x^{m - 1} \cos a x$}}:
:$(1): \quad \ds \int x^{m - 1} \cos a x \rd x = \frac {x^{m - 1} \sin a x} a - \frac {m - 1} a \int x^{m - 2} \sin a x \rd x$
So:
{{... | :$\ds \int x^m \sin a x \rd x = \frac {-x^m \cos a x} a + \frac {m x^{m - 1} \sin a x} {a^2} - \frac {m \paren {m - 1} } {a^2} \int x^{m - 2} \sin a x \rd x$ | === [[Primitive of Power of x by Sine of a x/Lemma|Lemma]] ===
{{:Primitive of Power of x by Sine of a x/Lemma}}{{qed|lemma}}
From {{Lemma|Primitive of Power of x by Cosine of a x|proof = yes|disp = Primitive of $x^{m - 1} \cos a x$}}:
:$(1): \quad \ds \int x^{m - 1} \cos a x \rd x = \frac {x^{m - 1} \sin a x} a - \... | Primitive of Power of x by Sine of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Sine_of_a_x | [
"Primitive of Power of x by Sine of a x",
"Primitives involving Sine Function"
] | [] | [
"Primitive of Power of x by Sine of a x/Lemma"
] |
proofwiki-9540 | Primitive of Sine of a x over Power of x | :$\ds \int \frac {\sin a x} {x^n} \rd x = \frac {-\sin a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\cos a x} {x^{n - 1} } \rd x$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\rd v} {\rd x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sin a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cos a x
| c = Derivative of $\sin a x$... | :$\ds \int \frac {\sin a x} {x^n} \rd x = \frac {-\sin a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\cos a x} {x^{n - 1} } \rd x$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\rd v} {\rd x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sin a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cos a x
| c... | Primitive of Sine of a x over Power of x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_Power_of_x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_Power_of_x | [
"Primitives involving Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Sine Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9541 | Primitive of Reciprocal of Power of Sine of a x | :$\ds \int \frac {\d x} {\sin^n a x} = \frac {-\cos a x} {a \paren {n - 1} \sin^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\sin^{n - 2} a x}$ | 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 1 {\sin^{n - 2} a x}
| c =
}}
{{eqn | r = \sin^{- n + 2} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x... | :$\ds \int \frac {\d x} {\sin^n a x} = \frac {-\cos a x} {a \paren {n - 1} \sin^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\sin^{n - 2} a x}$ | 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 1 {\sin^{n - 2} a x}
| c =
}}
{{eqn | r = \sin^{- n + 2} a x
| c =
}}
{{eqn | ll= \leadsto
... | Primitive of Reciprocal of Power of Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Sine Function/Corollary",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Cosecant is Reciprocal of Sine",
"Primitive of Square of Cosecant of a x",
"Integration by Parts",
"Cotangent is Cosine divided by Sine",
"Sum of Squares of Sine and Co... |
proofwiki-9542 | Primitive of x over Power of Sine of a x | :$\ds \int \frac {x \rd x} {\sin^n a x} = \frac {-x \cos a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sin^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {x \rd x} {\sin^{n - 2} a x}$ | 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 x {\sin^{n - 2} a x}
| c =
}}
{{eqn | r = x \sin^{-n + 2} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d ... | :$\ds \int \frac {x \rd x} {\sin^n a x} = \frac {-x \cos a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sin^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {x \rd x} {\sin^{n - 2} a x}$ | 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 x {\sin^{n - 2} a x}
| c =
}}
{{eqn | r = x \sin^{-n + 2} a x
| c =
}}
{{eqn | ll= \leadsto
... | Primitive of x over Power of Sine of a x | https://proofwiki.org/wiki/Primitive_of_x_over_Power_of_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_over_Power_of_Sine_of_a_x | [
"Primitives involving Sine Function"
] | [] | [
"Definition:Primitive",
"Product Rule for Derivatives",
"Derivative of Sine Function/Corollary",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Cosecant is Reciprocal of Sine",
"Primitive of Square of Cosecant of a x",
"Integration by Parts",
"Linear Combination of Integrals/Inde... |
proofwiki-9543 | Primitive of x by Cosine of a x | :$\ds \int x \cos a x \rd x = \frac {\cos a x} {a^2} + \frac {x \sin a x} a + C$ | 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... | :$\ds \int x \cos a x \rd x = \frac {\cos a x} {a^2} + \frac {x \sin a x} a + C$ | 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 Cosine of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Cosine Function/Corollary",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Sine Function/Corollary"
] |
proofwiki-9544 | Primitive of x squared by Cosine of a x | :$\ds \int x^2 \cos a x \rd x = \frac {2 x \cos a x} {a^2} + \paren {\frac {x^2} a - \frac 2 {a^3} } \sin a x + C$ | 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^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = Derivative of Power
}}
{{end-eqn}}
and l... | :$\ds \int x^2 \cos a x \rd x = \frac {2 x \cos a x} {a^2} + \paren {\frac {x^2} a - \frac 2 {a^3} } \sin a x + C$ | 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^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = [[Derivative o... | Primitive of x squared by Cosine of a x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Primitive of Cosine Function/Corollary",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of x by Sine of a x"
] |
proofwiki-9545 | Primitive of x cubed by Cosine of a x | :$\ds \int x^3 \map \cos {a x} \rd x = \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \cos a x + \paren {\frac {x^3} a - \frac {6 x} {a^3} } \sin a x + C$ | 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^3
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 3 x^2
| c = Derivative of Power
}}
{{end-eqn}}
and... | :$\ds \int x^3 \map \cos {a x} \rd x = \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \cos a x + \paren {\frac {x^3} a - \frac {6 x} {a^3} } \sin a x + C$ | 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^3
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 3 x^2
| c = [[Derivative... | Primitive of x cubed by Cosine of a x | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Primitive of Cosine Function/Corollary",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of x squared by Sine of a x"
] |
proofwiki-9546 | Primitive of Cosine of a x over x squared | :$\ds \int \frac {\cos a x \rd x} {x^2} = \frac {-\cos a x} x - a \int \frac {\sin a x \rd x} x$ | 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 a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -a \sin a x
| c = Derivative of $\cos a x$
}}... | :$\ds \int \frac {\cos a x \rd x} {x^2} = \frac {-\cos a x} x - a \int \frac {\sin a x \rd x} x$ | 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 a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -a \sin a x
| c = [... | Primitive of Cosine of a x over x squared | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_x_squared | [
"Primitives involving Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Cosine Function/Corollary",
"Primitive of Power",
"Integration by Parts"
] |
proofwiki-9547 | Primitive of Reciprocal of Cosine of a x/Logarithm of Secant plus Tangent Form | :$\ds \int \frac {\d x} {\cos a x} = \frac 1 a \ln \size {\sec a x + \tan a z}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cos x}
| r = \int \sec x \rd x
| c = {{Defof|Real Secant Function}}
}}
{{eqn | r = \ln \size {\sec x + \tan x} + C
| c = Primitive of $\sec x$: Secant plus Tangent Form
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\cos a x}
| r = \frac 1 a ... | :$\ds \int \frac {\d x} {\cos a x} = \frac 1 a \ln \size {\sec a x + \tan a z}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cos x}
| r = \int \sec x \rd x
| c = {{Defof|Real Secant Function}}
}}
{{eqn | r = \ln \size {\sec x + \tan x} + C
| c = [[Primitive of Secant Function/Secant plus Tangent Form|Primitive of $\sec x$: Secant plus Tangent Form]]
}}
{{eqn | ll= \leadsto
... | Primitive of Reciprocal of Cosine of a x/Logarithm of Secant plus Tangent Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cosine_of_a_x/Logarithm_of_Secant_plus_Tangent_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cosine_of_a_x/Logarithm_of_Secant_plus_Tangent_Form | [
"Primitives involving Cosine Function"
] | [] | [
"Primitive of Secant Function/Secant plus Tangent Form",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9548 | Primitive of Reciprocal of Cosine of a x/Logarithm of Tangent Form | :$\ds \int \frac {\d x} {\cos a x} = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cos x}
| r = \int \sec x \rd x
| c = {{Defof|Real Secant Function}}
}}
{{eqn | r = \ln \size {\map \tan {\frac \pi 4 + \frac x 2} } + C
| c = Primitive of $\sec x$: Tangent plus Angle Form
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\cos a x}
... | :$\ds \int \frac {\d x} {\cos a x} = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cos x}
| r = \int \sec x \rd x
| c = {{Defof|Real Secant Function}}
}}
{{eqn | r = \ln \size {\map \tan {\frac \pi 4 + \frac x 2} } + C
| c = [[Primitive of Secant Function/Tangent plus Angle Form|Primitive of $\sec x$: Tangent plus Angle Form]]
}}
{{eqn |... | Primitive of Reciprocal of Cosine of a x/Logarithm of Tangent Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cosine_of_a_x/Logarithm_of_Tangent_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cosine_of_a_x/Logarithm_of_Tangent_Form | [
"Primitives involving Cosine Function"
] | [] | [
"Primitive of Secant Function/Tangent plus Angle Form",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9549 | Primitive of x over Cosine of a x | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\cos a x}
| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_n \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C
| c =
}}
{{eqn | r = \dfrac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 + \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144}... | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x}{\cos a x}
| r = \int x \sec a x \rd x
| c = {{Defof|Secant Function}}
}}
{{eqn | r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_n \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C
| c = Primitive of $x \sec a x$
}}
{{end-eqn}}
{{qed}} | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\cos a x}
| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_n \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C
| c =
}}
{{eqn | r = \dfrac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 + \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144}... | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x}{\cos a x}
| r = \int x \sec a x \rd x
| c = {{Defof|Secant Function}}
}}
{{eqn | r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_n \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C
| c = [[Primitive of x by Secant of a x|Primitive of $x \sec a x$]]... | Primitive of x over Cosine of a x | https://proofwiki.org/wiki/Primitive_of_x_over_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_over_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Primitive of x by Secant of a x"
] |
proofwiki-9550 | Primitive of Square of Cosine of a x | :$\ds \int \cos^2 a x \rd x = \frac x 2 + \frac {\sin 2 a x} {4 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \cos^2 x \rd x
| r = \frac x 2 + \frac {\sin 2 x} 4 + C
| c = Primitive of $\cos^2 x$
}}
{{eqn | ll= \leadsto
| l = \int \cos^2 a x \rd x
| r = \frac 1 a \paren {\frac {a x} 2 + \frac {\sin 2 a x} 4} + C
| c = Primitive of Function of Constant Multiple
}}
{{e... | :$\ds \int \cos^2 a x \rd x = \frac x 2 + \frac {\sin 2 a x} {4 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \cos^2 x \rd x
| r = \frac x 2 + \frac {\sin 2 x} 4 + C
| c = [[Primitive of Square of Cosine Function|Primitive of $\cos^2 x$]]
}}
{{eqn | ll= \leadsto
| l = \int \cos^2 a x \rd x
| r = \frac 1 a \paren {\frac {a x} 2 + \frac {\sin 2 a x} 4} + C
| c = [[Prim... | Primitive of Square of Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Primitive of Square of Cosine Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9551 | Primitive of x by Square of Cosine of a x | :$\ds \int x \cos^2 a x \rd x = \frac {x^2} 4 + \frac {x \sin 2 a x} {4 a} + \frac {\cos 2 a x} {8 a^2} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int x \cos^2 a x \rd x = \frac {x^2} 4 + \frac {x \sin 2 a x} {4 a} + \frac {\cos 2 a x} {8 a^2} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x by Square of Cosine of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Square of Cosine of a x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Sine Function/Corollary"
] |
proofwiki-9552 | Primitive of Cube of Cosine of a x | :$\ds \int \cos^3 a x \rd x = \frac {\sin a x} a - \frac {\sin^3 a x} {3 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \cos^3 a x \rd x
| r = \int \paren {\frac {3 \cos a x + \cos 3 a x} 4} \rd x
| c = Power Reduction Formula for $\cos^3$
}}
{{eqn | r = \frac 3 4 \int \cos a x \rd x + \frac 1 4 \int \cos 3 a x \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = \frac 3 4 \paren {... | :$\ds \int \cos^3 a x \rd x = \frac {\sin a x} a - \frac {\sin^3 a x} {3 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \cos^3 a x \rd x
| r = \int \paren {\frac {3 \cos a x + \cos 3 a x} 4} \rd x
| c = [[Power Reduction Formula for Cube of Cosine|Power Reduction Formula for $\cos^3$]]
}}
{{eqn | r = \frac 3 4 \int \cos a x \rd x + \frac 1 4 \int \cos 3 a x \rd x
| c = [[Linear Combinatio... | Primitive of Cube of Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Power Reduction Formulas/Cosine Cubed",
"Linear Combination of Integrals/Indefinite",
"Primitive of Cosine Function/Corollary",
"Triple Angle Formulas/Sine"
] |
proofwiki-9553 | Primitive of Fourth Power of Cosine of a x | :$\ds \int \cos^4 a x \rd x = \frac {3 x} 8 + \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \cos^4 a x \rd x
| r = \int \paren {\frac {3 + 4 \cos 2 x + \cos 4 x} 8} \rd x
| c = Power Reduction Formula for $\cos^4$
}}
{{eqn | r = \frac 3 8 \int \d x + \frac 1 2 \int \cos 2 a x \rd x + \frac 1 8 \int \cos 4 a x \rd x
| c = Linear Combination of Primitives
}}
{{eq... | :$\ds \int \cos^4 a x \rd x = \frac {3 x} 8 + \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \cos^4 a x \rd x
| r = \int \paren {\frac {3 + 4 \cos 2 x + \cos 4 x} 8} \rd x
| c = [[Power Reduction Formula for 4th Power of Cosine|Power Reduction Formula for $\cos^4$]]
}}
{{eqn | r = \frac 3 8 \int \d x + \frac 1 2 \int \cos 2 a x \rd x + \frac 1 8 \int \cos 4 a x \rd x
... | Primitive of Fourth Power of Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Fourth_Power_of_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Fourth_Power_of_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Power Reduction Formulas/Cosine to 4th",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Cosine Function/Corollary"
] |
proofwiki-9554 | Primitive of Reciprocal of Square of Cosine of a x | :$\ds \int \frac {\d x} {\cos^2 a x} = \frac {\tan a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cos^2 a x}
| r = \int \sec^2 a x \rd x
| c = {{Defof|Cosecant|subdef = Analysis}}
}}
{{eqn | r = \frac {\tan a x} a + C
| c = Primitive of $\sec^2 a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {\cos^2 a x} = \frac {\tan a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cos^2 a x}
| r = \int \sec^2 a x \rd x
| c = {{Defof|Cosecant|subdef = Analysis}}
}}
{{eqn | r = \frac {\tan a x} a + C
| c = [[Primitive of Square of Secant of a x|Primitive of $\sec^2 a x$]]
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of Square of Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Primitive of Square of Secant of a x"
] |
proofwiki-9555 | Primitive of Square of Secant of a x | :$\ds \int \sec^2 a x \rd x = \frac {\tan a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \sec^2 x \rd x
| r = \tan x + C
| c = Primitive of $\sec^2 x$
}}
{{eqn | ll= \leadsto
| l = \int \sec^2 a x \rd x
| r = \frac 1 a \paren {\tan a x} + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\tan a x} a + C
| c = simplify... | :$\ds \int \sec^2 a x \rd x = \frac {\tan a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \sec^2 x \rd x
| r = \tan x + C
| c = [[Primitive of Square of Secant Function|Primitive of $\sec^2 x$]]
}}
{{eqn | ll= \leadsto
| l = \int \sec^2 a x \rd x
| r = \frac 1 a \paren {\tan a x} + C
| c = [[Primitive of Function of Constant Multiple]]
}}
{{eqn | ... | Primitive of Square of Secant of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Secant_of_a_x | [
"Primitives involving Secant Function"
] | [] | [
"Primitive of Square of Secant Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9556 | Primitive of Secant of a x/Tangent Form | :$\ds \int \sec a x \rd x = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$
where $\map \tan {\dfrac \pi 4 + \dfrac {a x} 2} \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \sec x \rd x
| r = \ln \size {\map \tan {\frac \pi 4 + \frac x 2} }
| c = Primitive of $\sec x$: Tangent plus Angle Form
}}
{{eqn | ll= \leadsto
| l = \int \sec a x \rd x
| r = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C
| c = Primitive... | :$\ds \int \sec a x \rd x = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$
where $\map \tan {\dfrac \pi 4 + \dfrac {a x} 2} \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \sec x \rd x
| r = \ln \size {\map \tan {\frac \pi 4 + \frac x 2} }
| c = [[Primitive of Secant Function/Tangent plus Angle Form|Primitive of $\sec x$: Tangent plus Angle Form]]
}}
{{eqn | ll= \leadsto
| l = \int \sec a x \rd x
| r = \frac 1 a \ln \size {\map \tan ... | Primitive of Secant of a x/Tangent Form | https://proofwiki.org/wiki/Primitive_of_Secant_of_a_x/Tangent_Form | https://proofwiki.org/wiki/Primitive_of_Secant_of_a_x/Tangent_Form | [
"Primitive of Secant of a x"
] | [] | [
"Primitive of Secant Function/Tangent plus Angle Form",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9557 | Primitive of Secant of a x/Secant plus Tangent Form | :$\ds \int \sec a x \rd x = \frac 1 a \ln \size {\sec a x + \tan a x} + C$
where $\sec a x + \tan a x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \sec x \rd x
| r = \ln \size {\sec x + \tan x}
| c = Primitive of $\sec x$: Secant plus Tangent Form
}}
{{eqn | ll= \leadsto
| l = \int \sec a x \rd x
| r = \frac 1 a \ln \size {\sec a x + \tan a x} + C
| c = Primitive of Function of Constant Multiple
}}
{{en... | :$\ds \int \sec a x \rd x = \frac 1 a \ln \size {\sec a x + \tan a x} + C$
where $\sec a x + \tan a x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \sec x \rd x
| r = \ln \size {\sec x + \tan x}
| c = [[Primitive of Secant Function/Secant plus Tangent Form|Primitive of $\sec x$: Secant plus Tangent Form]]
}}
{{eqn | ll= \leadsto
| l = \int \sec a x \rd x
| r = \frac 1 a \ln \size {\sec a x + \tan a x} + C
... | Primitive of Secant of a x/Secant plus Tangent Form | https://proofwiki.org/wiki/Primitive_of_Secant_of_a_x/Secant_plus_Tangent_Form | https://proofwiki.org/wiki/Primitive_of_Secant_of_a_x/Secant_plus_Tangent_Form | [
"Primitive of Secant of a x"
] | [] | [
"Primitive of Secant Function/Secant plus Tangent Form",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9558 | Primitive of Power of Secant of a x | :$\ds \int \sec^n a x \rd x = \frac {\sec^{n - 2} a x \tan a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sec^{n - 2} a x \rd x$
where $n \ne 1$. | 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^{n - 2} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \paren {n - 2} \sec^{n - 3} a x \sec a x \... | :$\ds \int \sec^n a x \rd x = \frac {\sec^{n - 2} a x \tan a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sec^{n - 2} a x \rd x$
where $n \ne 1$. | 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 = \sec^{n - 2} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \paren {n - 2} \... | Primitive of Power of Secant of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Secant_of_a_x | [
"Primitives involving Secant Function"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Derivative of Secant Function",
"Derivative of Composite 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"
] |
proofwiki-9559 | Primitive of Reciprocal of Cube of Cosine of a x | :$\ds \int \frac {\d x} {\cos^3 a x} = \frac {\sin a x} {2 a \cos^2 a x} + \frac 1 {2 a} \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cos^3 a x}
| r = \int \sec^3 a x \rd x
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac {\sec a x \tan a x} {2 a} + \frac 1 2 \int \sec a x \rd x
| c = Primitive of $\sec^n a x$
}}
{{eqn | r = \frac {\sin a x} {2 a \cos^2 a x} + \frac 1 2 \int \se... | :$\ds \int \frac {\d x} {\cos^3 a x} = \frac {\sin a x} {2 a \cos^2 a x} + \frac 1 {2 a} \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cos^3 a x}
| r = \int \sec^3 a x \rd x
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | 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 $\sec^n a x$]]
}}
{{eqn | r = \frac {\si... | Primitive of Reciprocal of Cube of Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cube_of_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cube_of_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Secant is Reciprocal of Cosine",
"Primitive of Power of Secant of a x",
"Secant is Reciprocal of Cosine",
"Tangent is Sine divided by Cosine",
"Primitive of Secant of a x/Tangent Form"
] |
proofwiki-9560 | Primitive of Cosine of a x by Cosine of b x | :$\ds \int \cos a x \cos b x \rd x = \frac {\map \sin {\paren {a - b} x} } {2 \paren {a - b} } + \frac {\map \sin {\paren {a + b} x} } {2 \paren {a + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \cos a x \cos b x \rd x
| r = \int \paren {\frac {\map \cos {a x - b x} + \map \cos {a x + b x} } 2} \rd x
| c = Werner Formula for Cosine by Cosine
}}
{{eqn | r = \frac 1 2 \int \map \cos {\paren {a - b} x} \rd x + \frac 1 2 \int \map \cos {\paren {a + b} x} \rd x
| c =... | :$\ds \int \cos a x \cos b x \rd x = \frac {\map \sin {\paren {a - b} x} } {2 \paren {a - b} } + \frac {\map \sin {\paren {a + b} x} } {2 \paren {a + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \cos a x \cos b x \rd x
| r = \int \paren {\frac {\map \cos {a x - b x} + \map \cos {a x + b x} } 2} \rd x
| c = [[Werner Formula for Cosine by Cosine]]
}}
{{eqn | r = \frac 1 2 \int \map \cos {\paren {a - b} x} \rd x + \frac 1 2 \int \map \cos {\paren {a + b} x} \rd x
|... | Primitive of Cosine of a x by Cosine of b x | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_by_Cosine_of_b_x | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_by_Cosine_of_b_x | [
"Primitives involving Cosine Function"
] | [] | [
"Werner Formulas/Cosine by Cosine",
"Linear Combination of Integrals/Indefinite",
"Primitive of Cosine Function/Corollary"
] |
proofwiki-9561 | Primitive of Reciprocal of 1 minus Cosine of x/Corollary | :$\ds \int \frac {\d x} {1 - \cos a x} = -\frac 1 a \cot \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {1 - \cos x}
| r = -\cot \frac x 2 + C
| c = Primitive of $\dfrac 1 {1 - \cos x}$
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {1 - \cos a x}
| r = \frac {-1} a \cot \frac {a x} 2 + C
| c = Primitive of Function of Constant Multiple
}}
{{end-e... | :$\ds \int \frac {\d x} {1 - \cos a x} = -\frac 1 a \cot \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {1 - \cos x}
| r = -\cot \frac x 2 + C
| c = [[Primitive of Reciprocal of 1 minus Cosine of x|Primitive of $\dfrac 1 {1 - \cos x}$]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {1 - \cos a x}
| r = \frac {-1} a \cot \frac {a x} 2 + C
| c = [[... | Primitive of Reciprocal of 1 minus Cosine of x/Corollary | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_minus_Cosine_of_x/Corollary | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_minus_Cosine_of_x/Corollary | [
"Primitive of Reciprocal of 1 minus Cosine of x"
] | [] | [
"Primitive of Reciprocal of 1 minus Cosine of x",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9562 | Primitive of x over 1 minus Cosine of a x | :$\ds \int \frac {x \rd x} {1 - \cos a x} = \frac {-x} a \cot \frac {a x} 2 + \frac 2 {a^2} \ln \size {\sin \frac {a x} 2} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Primitive of Power
}}
{{end-eqn}}
and let:
{... | :$\ds \int \frac {x \rd x} {1 - \cos a x} = \frac {-x} a \cot \frac {a x} 2 + \frac 2 {a^2} \ln \size {\sin \frac {a x} 2} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Primitive of P... | Primitive of x over 1 minus Cosine of a x | https://proofwiki.org/wiki/Primitive_of_x_over_1_minus_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_over_1_minus_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Definition:Primitive",
"Primitive of Power",
"Primitive of Reciprocal of 1 minus Cosine of x/Corollary",
"Integration by Parts",
"Primitive of Cotangent of a x"
] |
proofwiki-9563 | Primitive of Reciprocal of 1 plus Cosine of x/Corollary | :$\ds \int \frac {\d x} {1 + \cos a x} = \frac 1 a \tan \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {1 + \cos x}
| r = \tan \frac x 2 + C
| c = Primitive of $\dfrac 1 {1 + \cos x}$
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {1 + \cos a x}
| r = \frac 1 a \tan \frac {a x} 2 + C
| c = Primitive of Function of Constant Multiple
}}
{{end-eqn}}... | :$\ds \int \frac {\d x} {1 + \cos a x} = \frac 1 a \tan \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {1 + \cos x}
| r = \tan \frac x 2 + C
| c = [[Primitive of Reciprocal of 1 plus Cosine of x|Primitive of $\dfrac 1 {1 + \cos x}$]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {1 + \cos a x}
| r = \frac 1 a \tan \frac {a x} 2 + C
| c = [[Primi... | Primitive of Reciprocal of 1 plus Cosine of x/Corollary | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_plus_Cosine_of_x/Corollary | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_plus_Cosine_of_x/Corollary | [
"Primitive of Reciprocal of 1 plus Cosine of x"
] | [] | [
"Primitive of Reciprocal of 1 plus Cosine of x",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9564 | Primitive of x over 1 plus Cosine of a x | :$\ds \int \frac {x \rd x} {1 + \cos a x} = \frac x a \tan \frac {a x} 2 + \frac 2 {a^2} \ln \size {\cos \frac {a x} 2} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Primitive of Power
}}
{{end-eqn}}
and let:
{... | :$\ds \int \frac {x \rd x} {1 + \cos a x} = \frac x a \tan \frac {a x} 2 + \frac 2 {a^2} \ln \size {\cos \frac {a x} 2} + C$ | 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 = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Primitive of Pow... | Primitive of x over 1 plus Cosine of a x | https://proofwiki.org/wiki/Primitive_of_x_over_1_plus_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_over_1_plus_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Definition:Primitive",
"Primitive of Power",
"Primitive of Reciprocal of 1 plus Cosine of x/Corollary",
"Integration by Parts",
"Primitive of Tangent of a x/Cosine Form"
] |
proofwiki-9565 | Primitive of Reciprocal of Square of 1 minus Cosine of a x | :$\ds \int \frac {\d x} {\paren {1 - \cos a x}^2} = \frac {-1} {2 a} \cot \frac {a x} 2 - \frac 1 {6 a} \cot^3 \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {1 - \cos a x}^2}
| r = \int \paren {\frac 1 2 \csc^2 \frac {a x} 2}^2 \rd x
| c = Reciprocal of One Minus Cosine
}}
{{eqn | r = \frac 1 4 \int \csc^4 \frac {a x} 2 \rd x
| c = simplifying
}}
{{eqn | r = \frac 1 4 \paren {\frac{-\csc^2 \dfrac {a x} 2... | :$\ds \int \frac {\d x} {\paren {1 - \cos a x}^2} = \frac {-1} {2 a} \cot \frac {a x} 2 - \frac 1 {6 a} \cot^3 \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {1 - \cos a x}^2}
| r = \int \paren {\frac 1 2 \csc^2 \frac {a x} 2}^2 \rd x
| c = [[Reciprocal of One Minus Cosine]]
}}
{{eqn | r = \frac 1 4 \int \csc^4 \frac {a x} 2 \rd x
| c = simplifying
}}
{{eqn | r = \frac 1 4 \paren {\frac{-\csc^2 \dfrac {a ... | Primitive of Reciprocal of Square of 1 minus Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_1_minus_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_1_minus_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Reciprocal of One Minus Cosine",
"Primitive of Power of Cosecant of a x",
"Primitive of Square of Cosecant of a x",
"Sum of Squares of Sine and Cosine/Corollary 2"
] |
proofwiki-9566 | Primitive of Reciprocal of Square of 1 plus Cosine of a x | :$\ds \int \frac {\d x} {\paren {1 + \cos a x}^2} = \frac 1 {2 a} \tan \frac {a x} 2 + \frac 1 {6 a} \tan^3 \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {1 + \cos a x}^2}
| r = \int \paren {\frac 1 2 \sec^2 \frac {a x} 2}^2 \rd x
| c = Reciprocal of One Plus Cosine
}}
{{eqn | r = \frac 1 4 \int \sec^4 \frac {a x} 2 \rd x
| c = simplifying
}}
{{eqn | r = \frac 1 4 \paren {\frac {\sec^2 \dfrac {a x} 2 ... | :$\ds \int \frac {\d x} {\paren {1 + \cos a x}^2} = \frac 1 {2 a} \tan \frac {a x} 2 + \frac 1 {6 a} \tan^3 \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {1 + \cos a x}^2}
| r = \int \paren {\frac 1 2 \sec^2 \frac {a x} 2}^2 \rd x
| c = [[Reciprocal of One Plus Cosine]]
}}
{{eqn | r = \frac 1 4 \int \sec^4 \frac {a x} 2 \rd x
| c = simplifying
}}
{{eqn | r = \frac 1 4 \paren {\frac {\sec^2 \dfrac {a x... | Primitive of Reciprocal of Square of 1 plus Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_1_plus_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_1_plus_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Reciprocal of One Plus Cosine",
"Primitive of Power of Secant of a x",
"Primitive of Square of Secant of a x",
"Sum of Squares of Sine and Cosine/Corollary 1"
] |
proofwiki-9567 | Primitive of Reciprocal of p plus q by Cosine of a x | :<nowiki>$\ds \int \frac {\rd x} {p + q \cos a x} = \begin {cases}
\dfrac 2 {a \sqrt {p^2 - q^2} } \map \arctan {\sqrt {\dfrac {p - q} {p + q} } \tan \dfrac {a x} 2} + C & : p^2 > q^2 \\ \\
\dfrac 1 {a \sqrt {q^2 - p^2} } \ln \size {\dfrac {\tan \dfrac {a x} 2 + \sqrt {\dfrac {q + p} {q - p} } } {\tan \dfrac {a x} 2 -... | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \cos a x}
| r = \frac 2 {a \paren {p - q} } \int \frac {\d u} {u^2 + \dfrac {p + q} {p - q} }
| c = Weierstrass Substitution: $u = \tan \dfrac {a x} 2$
}}
{{end-eqn}}
Let $p^2 > q^2$.
Then, by Sign of Quotient of Factors of Difference of Squares:
:$\dfrac {... | :<nowiki>$\ds \int \frac {\rd x} {p + q \cos a x} = \begin {cases}
\dfrac 2 {a \sqrt {p^2 - q^2} } \map \arctan {\sqrt {\dfrac {p - q} {p + q} } \tan \dfrac {a x} 2} + C & : p^2 > q^2 \\ \\
\dfrac 1 {a \sqrt {q^2 - p^2} } \ln \size {\dfrac {\tan \dfrac {a x} 2 + \sqrt {\dfrac {q + p} {q - p} } } {\tan \dfrac {a x} 2 -... | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \cos a x}
| r = \frac 2 {a \paren {p - q} } \int \frac {\d u} {u^2 + \dfrac {p + q} {p - q} }
| c = [[Primitive of Reciprocal of p plus q by Cosine of a x/Weierstrass Substitution|Weierstrass Substitution]]: $u = \tan \dfrac {a x} 2$
}}
{{end-eqn}}
Let $p... | Primitive of Reciprocal of p plus q by Cosine of a x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Cosine_of_a_x/Proof_1 | [
"Primitive of Reciprocal of p plus q by Cosine of a x",
"Primitives involving Cosine Function"
] | [] | [
"Primitive of Reciprocal of p plus q by Cosine of a x/Weierstrass Substitution",
"Sign of Quotient of Factors of Difference of Squares",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Sign of Quotient of Factors of Difference of Squares",
"Primitive of Reciprocal of x squared minus ... |
proofwiki-9568 | Primitive of Reciprocal of square of p plus q by Cosine of a x | :$\ds \int \frac {\d x} {\paren {p + q \cos a x}^2} = \frac {q \sin a x} {a \paren {q^2 - p^2} \paren {p + q \cos a x} } - \frac p {q^2 - p^2} \int \frac {\d x} {p + q \cos a x}$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\dfrac {\sin a x} {p + q \cos a x} }
| r = \dfrac {\paren {p + q \cos a x} \map {\frac \d {\d x} } {\sin a x} - \sin a x \map {\frac \d {\d x} } {p + q \sin a x} } {\paren {p + q \cos a x}^2}
| c = Quotient Rule for Derivatives
}}
{{eqn | r = \dfrac {\pare... | :$\ds \int \frac {\d x} {\paren {p + q \cos a x}^2} = \frac {q \sin a x} {a \paren {q^2 - p^2} \paren {p + q \cos a x} } - \frac p {q^2 - p^2} \int \frac {\d x} {p + q \cos a x}$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\dfrac {\sin a x} {p + q \cos a x} }
| r = \dfrac {\paren {p + q \cos a x} \map {\frac \d {\d x} } {\sin a x} - \sin a x \map {\frac \d {\d x} } {p + q \sin a x} } {\paren {p + q \cos a x}^2}
| c = [[Quotient Rule for Derivatives]]
}}
{{eqn | r = \dfrac {\... | Primitive of Reciprocal of square of p plus q by Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_square_of_p_plus_q_by_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_square_of_p_plus_q_by_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Quotient Rule for Derivatives",
"Derivative of Cosine Function",
"Derivative of Sine Function",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-9569 | Primitive of Reciprocal of p squared plus square of q by Cosine of a x | :$\ds \int \frac {\d x} {p^2 + q^2 \cos^2 a x} = \frac 1 {a p \sqrt{p^2 + q^2} } \arctan \frac {p \tan a x} {\sqrt {p^2 + q^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 + q^2 \cos^2 a x}
| r = \int \frac {\csc^2 a x \rd x} {p^2 \csc^2 a x + q^2 \cot^2 a x}
| c = multiplying the numerator and the denominator by $\csc^2 a x$
}}
{{eqn | r = \int \frac {\csc^2 a x \rd x} {p^2 + \paren {p^2 + q^2} \cot^2 a x}
| c = Differen... | :$\ds \int \frac {\d x} {p^2 + q^2 \cos^2 a x} = \frac 1 {a p \sqrt{p^2 + q^2} } \arctan \frac {p \tan a x} {\sqrt {p^2 + q^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 + q^2 \cos^2 a x}
| r = \int \frac {\csc^2 a x \rd x} {p^2 \csc^2 a x + q^2 \cot^2 a x}
| c = multiplying the [[Definition:Numerator|numerator]] and the [[Definition:Denominator|denominator]] by $\csc^2 a x$
}}
{{eqn | r = \int \frac {\csc^2 a x \rd x} {p^2 +... | Primitive of Reciprocal of p squared plus square of q by Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_plus_square_of_q_by_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_plus_square_of_q_by_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sum of Squares of Sine and Cosine/Corollary 2",
"Derivative of Cotangent Function",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Arctangent of Reciprocal equals Arccotangent",
"Sum of Arctangent and Arccotang... |
proofwiki-9570 | Primitive of Reciprocal of p squared minus square of q by Cosine of a x | :$\ds \int \frac {\rd x} {p^2 - q^2 \cos^2 a x} = \begin {cases}
\dfrac 1 {a p \sqrt {p^2 - q^2} } \arctan \dfrac {p \tan a x} {\sqrt {p^2 - q^2} } & : p^2 > q^2 \\
\dfrac 1 {2 a p \sqrt {q^2 - p^2} } \ln \size {\dfrac {p \tan a x - \sqrt {q^2 - p^2} } {p \tan a x + \sqrt {q^2 - p^2} } } & : p^2 < q^2
\end {cases}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 - q^2 \cos^2 a x}
| r = \int \frac {\csc^2 a x \rd x} {p^2 \csc^2 a x - q^2 \cot^2 a x}
| c = multiplying numerator and denominator by $\csc^2 a x$
}}
{{eqn | r = \int \frac {\csc^2 a x \rd x} {p^2 \paren {1 + \cot^2 a x} - q^2 \cot^2 a x}
| c = Differe... | :$\ds \int \frac {\rd x} {p^2 - q^2 \cos^2 a x} = \begin {cases}
\dfrac 1 {a p \sqrt {p^2 - q^2} } \arctan \dfrac {p \tan a x} {\sqrt {p^2 - q^2} } & : p^2 > q^2 \\
\dfrac 1 {2 a p \sqrt {q^2 - p^2} } \ln \size {\dfrac {p \tan a x - \sqrt {q^2 - p^2} } {p \tan a x + \sqrt {q^2 - p^2} } } & : p^2 < q^2
\end {cases}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 - q^2 \cos^2 a x}
| r = \int \frac {\csc^2 a x \rd x} {p^2 \csc^2 a x - q^2 \cot^2 a x}
| c = multiplying [[Definition:Numerator|numerator]] and [[Definition:Denominator|denominator]] by $\csc^2 a x$
}}
{{eqn | r = \int \frac {\csc^2 a x \rd x} {p^2 \paren {1... | Primitive of Reciprocal of p squared minus square of q by Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_minus_square_of_q_by_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_minus_square_of_q_by_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sum of Squares of Sine and Cosine/Corollary 2",
"Derivative of Cotangent Function",
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Arctangent of Reciprocal equals Arccotangent",
... |
proofwiki-9571 | Primitive of Power of x by Cosine of a x | :$\ds \int x^m \cos a x \rd x = \frac {x^m \sin a x} a + \frac {m x^{m - 1} \cos a x} {a^2} - \frac {m \paren {m - 1} } {a^2} \int x^{m - 2} \cos a x \rd x$ | === Lemma ===
{{:Primitive of Power of x by Cosine of a x/Lemma}}{{qed|lemma}}
From {{Lemma|Primitive of Power of x by Sine of a x|proof = yes|disp = Primitive of $x^{m - 1} \sin a x$}}:
:$(1): \quad \ds \int x^{m - 1} \sin a x \rd x = \frac {-x^{m - 1} \cos a x} a + \frac {m - 1} a \int x^{m - 2} \cos a x \rd x$
So:
{... | :$\ds \int x^m \cos a x \rd x = \frac {x^m \sin a x} a + \frac {m x^{m - 1} \cos a x} {a^2} - \frac {m \paren {m - 1} } {a^2} \int x^{m - 2} \cos a x \rd x$ | === [[Primitive of Power of x by Cosine of a x/Lemma|Lemma]] ===
{{:Primitive of Power of x by Cosine of a x/Lemma}}{{qed|lemma}}
From {{Lemma|Primitive of Power of x by Sine of a x|proof = yes|disp = Primitive of $x^{m - 1} \sin a x$}}:
:$(1): \quad \ds \int x^{m - 1} \sin a x \rd x = \frac {-x^{m - 1} \cos a x} a ... | Primitive of Power of x by Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Cosine_of_a_x | [
"Primitive of Power of x by Cosine of a x",
"Primitives involving Cosine Function"
] | [] | [
"Primitive of Power of x by Cosine of a x/Lemma"
] |
proofwiki-9572 | Primitive of Cosine of a x over Power of x | :$\ds \int \frac {\cos a x} {x^n} \rd x = \frac {-\cos a x} {\paren {n - 1} x^{n - 1} } - \frac a {n - 1} \int \frac {\sin a x} {x^{n - 1} } \rd x$ | 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 a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -a \sin a x
| c = Derivative of $\cos a x$
}}... | :$\ds \int \frac {\cos a x} {x^n} \rd x = \frac {-\cos a x} {\paren {n - 1} x^{n - 1} } - \frac a {n - 1} \int \frac {\sin a x} {x^{n - 1} } \rd x$ | 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 a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -a \sin a x
| c = [... | Primitive of Cosine of a x over Power of x | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_Power_of_x | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_Power_of_x | [
"Primitives involving Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Cosine Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9573 | Primitive of Reciprocal of Power of Cosine of a x | :$\ds \int \frac {\d x} {\cos^n a x} = \frac {\sin a x} {a \paren {n - 1} \cos^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cos^{n - 2} a x}$ | 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 1 {\cos^{n - 2} a x}
| c =
}}
{{eqn | r = \cos^{- n + 2} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x... | :$\ds \int \frac {\d x} {\cos^n a x} = \frac {\sin a x} {a \paren {n - 1} \cos^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cos^{n - 2} a x}$ | 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 1 {\cos^{n - 2} a x}
| c =
}}
{{eqn | r = \cos^{- n + 2} a x
| c =
}}
{{eqn | ll= \leadsto
... | Primitive of Reciprocal of Power of Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Cosine Function/Corollary",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Secant is Reciprocal of Cosine",
"Primitive of Square of Secant of a x",
"Integration by Parts",
"Tangent is Sine divided by Cosine",
"Sum of Squares of Sine and Cosi... |
proofwiki-9574 | Primitive of x over Power of Cosine of a x | :$\ds \int \frac {x \rd x} {\cos^n a x} = \frac {x \sin a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \cos^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cos^{n - 2} a x} + C$ | 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 = \frac x {\cos^{n - 2} a x}
| c =
}}
{{eqn | r = x \cos^{- n + 2} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d... | :$\ds \int \frac {x \rd x} {\cos^n a x} = \frac {x \sin a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \cos^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cos^{n - 2} a x} + C$ | 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 = \frac x {\cos^{n - 2} a x}
| c =
}}
{{eqn | r = x \cos^{- n + 2} a x
| c =
}}
{{eqn | ll= \leadsto
... | Primitive of x over Power of Cosine of a x | https://proofwiki.org/wiki/Primitive_of_x_over_Power_of_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_over_Power_of_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Definition:Primitive",
"Product Rule for Derivatives",
"Derivative of Cosine Function/Corollary",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Secant is Reciprocal of Cosine",
"Primitive of Square of Secant of a x",
"Integration by Parts",
"Linear Combination of Integrals/Inde... |
proofwiki-9575 | Primitive of Sine of a x by Cosine of a x | :$\ds \int \sin a x \cos a x \rd x = \frac {\sin^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \sin a x \cos a x \rd x
| r = \int \frac {\sin 2 a x} 2 \rd x
| c = Double Angle Formula for Sine
}}
{{eqn | r = \frac {-\cos 2 a x} {4 a} + C
| c = Primitive of $\cos a x$
}}
{{eqn | r = \frac {-\paren {1 - 2 \sin^2 a x} } {4 a} + C
| c = {{Corollary|Double Angle ... | :$\ds \int \sin a x \cos a x \rd x = \frac {\sin^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \sin a x \cos a x \rd x
| r = \int \frac {\sin 2 a x} 2 \rd x
| c = [[Double Angle Formula for Sine]]
}}
{{eqn | r = \frac {-\cos 2 a x} {4 a} + C
| c = [[Primitive of Cosine of a x|Primitive of $\cos a x$]]
}}
{{eqn | r = \frac {-\paren {1 - 2 \sin^2 a x} } {4 a} + C
... | Primitive of Sine of a x by Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_by_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_by_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Double Angle Formulas/Sine",
"Primitive of Cosine Function/Corollary",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-9576 | Primitive of Sine of a x by Cosine of b x | :$\ds \int \sin a x \cos b x \rd x = \frac {-\map \cos {a - b} x} {2 \paren {a - b} } - \frac {\map \cos {a + b} x} {2 \paren {a + b} } + C$
for $a, b \in \R: a \ne b$ | {{begin-eqn}}
{{eqn | l = \int \sin a x \cos b x \rd x
| r = \int \paren {\dfrac {\map \sin {a x + b x} + \map \sin {a x - b x} } 2} \rd x
| c = Werner Formula for Sine by Cosine
}}
{{eqn | r = \frac 1 2 \int \map \sin {a - b} x \rd x + \frac 1 2 \int \map \sin {a + b} x \rd x
| c = Linear Combination... | :$\ds \int \sin a x \cos b x \rd x = \frac {-\map \cos {a - b} x} {2 \paren {a - b} } - \frac {\map \cos {a + b} x} {2 \paren {a + b} } + C$
for $a, b \in \R: a \ne b$ | {{begin-eqn}}
{{eqn | l = \int \sin a x \cos b x \rd x
| r = \int \paren {\dfrac {\map \sin {a x + b x} + \map \sin {a x - b x} } 2} \rd x
| c = [[Werner Formula for Sine by Cosine]]
}}
{{eqn | r = \frac 1 2 \int \map \sin {a - b} x \rd x + \frac 1 2 \int \map \sin {a + b} x \rd x
| c = [[Linear Combi... | Primitive of Sine of a x by Cosine of b x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_by_Cosine_of_b_x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_by_Cosine_of_b_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Werner Formulas/Sine by Cosine",
"Linear Combination of Integrals/Indefinite",
"Primitive of Cosine Function/Corollary"
] |
proofwiki-9577 | Primitive of Power of Sine of a x by Cosine of a x | :$\ds \int \sin^n a x \cos a x \rd x = \frac {\sin^{n + 1} a x} {\paren {n + 1} a} + C$
for $n \ne -1$. | {{begin-eqn}}
{{eqn | l = z
| r = \sin a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a \cos a x
| c = Primitive of $\sin a x$
}}
{{eqn | ll= \leadsto
| l = \int \sin^n a x \cos a x \rd x
| r = \int \frac {z^n \rd x} a
| c = Integration by Substitution... | :$\ds \int \sin^n a x \cos a x \rd x = \frac {\sin^{n + 1} a x} {\paren {n + 1} a} + C$
for $n \ne -1$. | {{begin-eqn}}
{{eqn | l = z
| r = \sin a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a \cos a x
| c = [[Primitive of Sine of a x|Primitive of $\sin a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \sin^n a x \cos a x \rd x
| r = \int \frac {z^n \rd x} a
| c ... | Primitive of Power of Sine of a x by Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Sine_of_a_x_by_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Sine_of_a_x_by_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Primitive of Sine Function/Corollary",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9578 | Primitive of Power of Cosine of a x by Sine of a x | :$\ds \int \cos^n a x \sin a x \rd x = \frac {-\cos^{n + 1} a x} {\paren {n + 1} a} + C$
for $n \ne -1$. | {{begin-eqn}}
{{eqn | l = z
| r = \cos a x
| c = Werner Formula for Sine by Cosine
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -a \sin a x
| c = Primitive of $\cos a x$
}}
{{eqn | ll= \leadsto
| l = \int \cos^n a x \sin a x \rd x
| r = \int \frac {-z^n \rd x} a
... | :$\ds \int \cos^n a x \sin a x \rd x = \frac {-\cos^{n + 1} a x} {\paren {n + 1} a} + C$
for $n \ne -1$. | {{begin-eqn}}
{{eqn | l = z
| r = \cos a x
| c = [[Werner Formula for Sine by Cosine]]
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -a \sin a x
| c = [[Primitive of Cosine of a x|Primitive of $\cos a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \cos^n a x \sin a x \rd x
|... | Primitive of Power of Cosine of a x by Sine of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Cosine_of_a_x_by_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Cosine_of_a_x_by_Sine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Werner Formulas/Sine by Cosine",
"Primitive of Cosine Function/Corollary",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9579 | Primitive of Sine of a x squared by Cosine of a x squared | :$\ds \int \sin^2 a x \cos^2 a x \rd x = \frac x 8 - \frac {\sin 4 a x} {32 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \sin^2 a x \cos^2 a x \rd x
| r = \int \sin^2 a x \paren {1 - \sin^2 a x} \rd x
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | r = \int \sin^2 a x \rd x - \int \sin^4 a x \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = \frac x 2 - \frac {\sin 2 a x} {4 a}... | :$\ds \int \sin^2 a x \cos^2 a x \rd x = \frac x 8 - \frac {\sin 4 a x} {32 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \sin^2 a x \cos^2 a x \rd x
| r = \int \sin^2 a x \paren {1 - \sin^2 a x} \rd x
| c = [[Sum of Squares of Sine and Cosine]]
}}
{{eqn | r = \int \sin^2 a x \rd x - \int \sin^4 a x \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = \frac x 2 - \frac {\sin 2 a ... | Primitive of Sine of a x squared by Cosine of a x squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_squared_by_Cosine_of_a_x_squared | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_squared_by_Cosine_of_a_x_squared/Proof_1 | [
"Primitives involving Sine Function and Cosine Function",
"Primitive of Sine of a x squared by Cosine of a x squared"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Linear Combination of Integrals/Indefinite",
"Primitive of Square of Sine of a x",
"Primitive of Fourth Power of Sine of a x"
] |
proofwiki-9580 | Primitive of Sine of a x squared by Cosine of a x squared | :$\ds \int \sin^2 a x \cos^2 a x \rd x = \frac x 8 - \frac {\sin 4 a x} {32 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \sin^2 a x \cos^2 a x \rd x
| r = \int \paren {\sin a x \cos a x}^2 \rd x
}}
{{eqn | r = \int \paren {\frac 1 2 \sin 2 a x}^2 \rd x
| c = Double Angle Formula for Sine
}}
{{eqn | r = \frac 1 4 \int \sin^2 2 a x \rd x
}}
{{eqn | r = \frac 1 4 \paren {\frac x 2 - \frac {\map \sin {2 x \ti... | :$\ds \int \sin^2 a x \cos^2 a x \rd x = \frac x 8 - \frac {\sin 4 a x} {32 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \sin^2 a x \cos^2 a x \rd x
| r = \int \paren {\sin a x \cos a x}^2 \rd x
}}
{{eqn | r = \int \paren {\frac 1 2 \sin 2 a x}^2 \rd x
| c = [[Double Angle Formula for Sine]]
}}
{{eqn | r = \frac 1 4 \int \sin^2 2 a x \rd x
}}
{{eqn | r = \frac 1 4 \paren {\frac x 2 - \frac {\map \sin {2 x... | Primitive of Sine of a x squared by Cosine of a x squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_squared_by_Cosine_of_a_x_squared | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_squared_by_Cosine_of_a_x_squared/Proof_2 | [
"Primitives involving Sine Function and Cosine Function",
"Primitive of Sine of a x squared by Cosine of a x squared"
] | [] | [
"Double Angle Formulas/Sine",
"Primitive of Square of Sine of a x"
] |
proofwiki-9581 | Primitive of Reciprocal of Sine of x by Cosine of x/Corollary | :$\ds \int \frac {\d x} {\sin a x \cos a x} = \frac 1 a \ln \size {\tan a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin x \cos x}
| r = \ln \size {\tan x} + C
| c = Primitive of $\dfrac 1 {\sin x \cos x}$
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\sin a x \cos a x}
| r = \frac 1 a \ln \size {\tan a x} + C
| c = Primitive of Function of Constant Multip... | :$\ds \int \frac {\d x} {\sin a x \cos a x} = \frac 1 a \ln \size {\tan a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin x \cos x}
| r = \ln \size {\tan x} + C
| c = [[Primitive of Reciprocal of Sine of x by Cosine of x|Primitive of $\dfrac 1 {\sin x \cos x}$]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\sin a x \cos a x}
| r = \frac 1 a \ln \size {\tan a x} ... | Primitive of Reciprocal of Sine of x by Cosine of x/Corollary | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_x_by_Cosine_of_x/Corollary | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_x_by_Cosine_of_x/Corollary | [
"Primitive of Reciprocal of Sine of x by Cosine of x"
] | [] | [
"Primitive of Reciprocal of Sine of x by Cosine of x",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9582 | Primitive of Square of Secant of a x over Tangent of a x | :$\ds \int \frac {\sec^2 a x \rd x} {\tan a x} = \frac 1 a \ln \size {\tan a x} + C$ | {{begin-eqn}}
{{eqn | l = \frac {\d} {\d x} \tan x
| r = \sec^2 x
| c = Derivative of Tangent Function
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sec^2 x \rd x} {\tan x}
| r = \ln \size {\tan a x} + C
| c = Primitive of Function under its Derivative
}}
{{eqn | ll= \leadsto
| l = \int... | :$\ds \int \frac {\sec^2 a x \rd x} {\tan a x} = \frac 1 a \ln \size {\tan a x} + C$ | {{begin-eqn}}
{{eqn | l = \frac {\d} {\d x} \tan x
| r = \sec^2 x
| c = [[Derivative of Tangent Function]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sec^2 x \rd x} {\tan x}
| r = \ln \size {\tan a x} + C
| c = [[Primitive of Function under its Derivative]]
}}
{{eqn | ll= \leadsto
| ... | Primitive of Square of Secant of a x over Tangent of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Secant_of_a_x_over_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Secant_of_a_x_over_Tangent_of_a_x | [
"Primitives involving Tangent Function",
"Primitives involving Secant Function"
] | [] | [
"Derivative of Tangent Function",
"Primitive of Function under its Derivative",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9583 | Primitive of Reciprocal of Square of Sine of a x by Cosine of a x | :$\ds \int \frac {\d x} {\sin^2 a x \cos a x} = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } - \frac 1 {a \sin a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin^2 a x \cos a x}
| r = \int \frac {\paren {\sin^2 a x + \cos^2 a x} \rd x} {\sin^2 a x \cos a x}
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | r = \int \frac {\sin^2 a x \rd x} {\sin^2 a x \cos a x} + \int \frac {\cos^2 a x \rd x} {\sin^2 a x \cos a x}
... | :$\ds \int \frac {\d x} {\sin^2 a x \cos a x} = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } - \frac 1 {a \sin a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin^2 a x \cos a x}
| r = \int \frac {\paren {\sin^2 a x + \cos^2 a x} \rd x} {\sin^2 a x \cos a x}
| c = [[Sum of Squares of Sine and Cosine]]
}}
{{eqn | r = \int \frac {\sin^2 a x \rd x} {\sin^2 a x \cos a x} + \int \frac {\cos^2 a x \rd x} {\sin^2 a x \cos a ... | Primitive of Reciprocal of Square of Sine of a x by Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Sine_of_a_x_by_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Sine_of_a_x_by_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Linear Combination of Integrals/Indefinite",
"Secant is Reciprocal of Cosine",
"Cotangent is Cosine divided by Sine",
"Cosecant is Reciprocal of Sine",
"Primitive of Secant of a x/Tangent Form",
"Primitive of Power of Cosecant of a x by Cotangent of a x"
] |
proofwiki-9584 | Primitive of Power of Cosecant of a x by Cotangent of a x | :$\ds \int \csc^n a x \cot a x \rd x = \frac {-\csc^n a x} {n a} + C$
for $n \ne 0$. | {{begin-eqn}}
{{eqn | l = z
| r = \csc a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -a \csc a x \cot a x
| c = Derivative of Cosecant Function, Chain Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \csc^n a x \cot a x \rd x
| r = \int \frac {-z^{n -... | :$\ds \int \csc^n a x \cot a x \rd x = \frac {-\csc^n a x} {n a} + C$
for $n \ne 0$. | {{begin-eqn}}
{{eqn | l = z
| r = \csc a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -a \csc a x \cot a x
| c = [[Derivative of Cosecant Function]], [[Chain Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \csc^n a x \cot a x \rd x
| r = \int \frac ... | Primitive of Power of Cosecant of a x by Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Cosecant_of_a_x_by_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Cosecant_of_a_x_by_Cotangent_of_a_x | [
"Primitives involving Cosecant Function",
"Primitives involving Cotangent Function"
] | [] | [
"Derivative of Cosecant Function",
"Derivative of Composite Function",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9585 | Primitive of Reciprocal of Sine of a x by Square of Cosine of a x | :$\ds \int \frac {\d x} {\sin a x \cos^2 a x} = \frac 1 a \ln \size {\tan \frac {a x} 2} + \frac 1 {a \cos a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x \cos^2 a x}
| r = \int \frac {\paren {\sin^2 a x + \cos^2 a x} \rd x} {\sin a x \cos^2 a x}
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | r = \int \frac {\sin^2 a x \rd x} {\sin a x \cos^2 a x} + \int \frac {\cos^2 a x \rd x} {\sin a x \cos^2 a x}
... | :$\ds \int \frac {\d x} {\sin a x \cos^2 a x} = \frac 1 a \ln \size {\tan \frac {a x} 2} + \frac 1 {a \cos a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x \cos^2 a x}
| r = \int \frac {\paren {\sin^2 a x + \cos^2 a x} \rd x} {\sin a x \cos^2 a x}
| c = [[Sum of Squares of Sine and Cosine]]
}}
{{eqn | r = \int \frac {\sin^2 a x \rd x} {\sin a x \cos^2 a x} + \int \frac {\cos^2 a x \rd x} {\sin a x \cos^2 a ... | Primitive of Reciprocal of Sine of a x by Square of Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x_by_Square_of_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x_by_Square_of_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Linear Combination of Integrals/Indefinite",
"Cosecant is Reciprocal of Sine",
"Tangent is Sine divided by Cosine",
"Secant is Reciprocal of Cosine",
"Primitive of Cosecant of a x/Tangent Form",
"Primitive of Power of Secant of a x by Tangent of a x"
] |
proofwiki-9586 | Primitive of Power of Secant of a x by Tangent of a x | :$\ds \int \sec^n a x \tan a x \rd x = \frac {\sec^n a x} {n a} + C$
for $n \ne 0$. | {{begin-eqn}}
{{eqn | l = z
| r = \sec a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a \sec a x \tan a x
| c = Derivative of Secant Function, Chain Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \sec^n a x \tan a x \rd x
| r = \int \frac {z^{n - 1} ... | :$\ds \int \sec^n a x \tan a x \rd x = \frac {\sec^n a x} {n a} + C$
for $n \ne 0$. | {{begin-eqn}}
{{eqn | l = z
| r = \sec a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a \sec a x \tan a x
| c = [[Derivative of Secant Function]], [[Chain Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \sec^n a x \tan a x \rd x
| r = \int \frac {z^... | Primitive of Power of Secant of a x by Tangent of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Secant_of_a_x_by_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Secant_of_a_x_by_Tangent_of_a_x | [
"Primitives involving Secant Function",
"Primitives involving Tangent Function"
] | [] | [
"Derivative of Secant Function",
"Derivative of Composite Function",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9587 | Primitive of Reciprocal of Square of Sine of a x by Square of Cosine of a x | :$\ds \int \frac {\d x} {\sin^2 a x \cos^2 a x} = \frac {-2 \cot 2 a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin^2 a x \cos^2 a x}
| r = \int \frac {\d x} {\left({\sin a x \cos a x}\right)^2}
| c =
}}
{{eqn | r = \int \frac {\d x} {\left({\frac {\sin 2 a x} 2}\right)^2}
| c = Double Angle Formula for Sine
}}
{{eqn | r = 4 \int \frac {\d x} {\sin^2 2 a x}
|... | :$\ds \int \frac {\d x} {\sin^2 a x \cos^2 a x} = \frac {-2 \cot 2 a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin^2 a x \cos^2 a x}
| r = \int \frac {\d x} {\left({\sin a x \cos a x}\right)^2}
| c =
}}
{{eqn | r = \int \frac {\d x} {\left({\frac {\sin 2 a x} 2}\right)^2}
| c = [[Double Angle Formula for Sine]]
}}
{{eqn | r = 4 \int \frac {\d x} {\sin^2 2 a x}
... | Primitive of Reciprocal of Square of Sine of a x by Square of Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Sine_of_a_x_by_Square_of_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Sine_of_a_x_by_Square_of_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Double Angle Formulas/Sine",
"Primitive of Constant Multiple of Function",
"Cosecant is Reciprocal of Sine",
"Primitive of Square of Cosecant of a x"
] |
proofwiki-9588 | Primitive of Square of Sine of a x over Cosine of a x | :$\ds \int \frac {\sin^2 a x \rd x} {\cos a x} = \frac {-\sin a x} a + \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\sin^2 a x \rd x} {\cos a x}
| r = \int \frac {\paren {1 - \cos^2 a x} \rd x} {\cos a x}
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | r = \int \frac {\d x} {\cos a x} - \int \cos a x \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = \int \sec a x \... | :$\ds \int \frac {\sin^2 a x \rd x} {\cos a x} = \frac {-\sin a x} a + \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\sin^2 a x \rd x} {\cos a x}
| r = \int \frac {\paren {1 - \cos^2 a x} \rd x} {\cos a x}
| c = [[Sum of Squares of Sine and Cosine]]
}}
{{eqn | r = \int \frac {\d x} {\cos a x} - \int \cos a x \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = \int \s... | Primitive of Square of Sine of a x over Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Sine_of_a_x_over_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Sine_of_a_x_over_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Linear Combination of Integrals/Indefinite",
"Secant is Reciprocal of Cosine",
"Primitive of Cosine Function/Corollary",
"Primitive of Secant of a x/Tangent Form"
] |
proofwiki-9589 | Primitive of Square of Cosine of a x over Sine of a x | :$\ds \int \frac {\cos^2 a x \rd x} {\sin a x} = \frac {\cos a x} a + \frac 1 a \ln \size {\tan \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\cos^2 a x \rd x} {\sin a x}
| r = \int \frac {\paren {1 - \sin^2 a x} \rd x} {\sin a x}
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | r = \int \frac {\d x} {\sin a x} - \int \sin a x \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = \int \csc a x \... | :$\ds \int \frac {\cos^2 a x \rd x} {\sin a x} = \frac {\cos a x} a + \frac 1 a \ln \size {\tan \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\cos^2 a x \rd x} {\sin a x}
| r = \int \frac {\paren {1 - \sin^2 a x} \rd x} {\sin a x}
| c = [[Sum of Squares of Sine and Cosine]]
}}
{{eqn | r = \int \frac {\d x} {\sin a x} - \int \sin a x \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = \int \c... | Primitive of Square of Cosine of a x over Sine of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosine_of_a_x_over_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Cosine_of_a_x_over_Sine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Linear Combination of Integrals/Indefinite",
"Cosecant is Reciprocal of Sine",
"Primitive of Sine Function/Corollary",
"Primitive of Cosecant of a x/Tangent Form"
] |
proofwiki-9590 | Primitive of Reciprocal of Cosine of a x by 1 plus Sine of a x | :$\ds \int \frac {\d x} {\cos a x \paren {1 + \sin a x} } = \frac {-1} {2 a \paren {1 + \sin a x} } + \frac 1 {2 a} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 4} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sin a x
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = a \cos a x
| c = Derivative of $\sin a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cos a x \paren {1 + \sin a x} }
| r = \int \frac {\cos a x \rd x} {\cos^2 a x \paren... | :$\ds \int \frac {\d x} {\cos a x \paren {1 + \sin a x} } = \frac {-1} {2 a \paren {1 + \sin a x} } + \frac 1 {2 a} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 4} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sin a x
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = a \cos a x
| c = [[Derivative of Sine of a x|Derivative of $\sin a x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cos a x \paren {1 + \sin a x} }
| r = \int \frac {\c... | Primitive of Reciprocal of Cosine of a x by 1 plus Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cosine_of_a_x_by_1_plus_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cosine_of_a_x_by_1_plus_Sine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Derivative of Sine Function/Corollary",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sum of Squares of Sine and Cosine",
"Integration by Substitution",
"Difference of Two Squares",
"Primitive of Reciprocal of a x + b squared by p x + q",
"Reciprocal of One Minus Sine",
"Reci... |
proofwiki-9591 | Primitive of Reciprocal of Cosine of a x by 1 minus Sine of a x | :$\ds \int \frac {\d x} {\cos a x \paren {1 - \sin a x} } = \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {2 a} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 4} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sin a x
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = a \cos a x
| c = Derivative of $\sin a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cos a x \paren {1 - \sin a x} }
| r = \int \frac {\cos a x \rd x} {\cos^2 a x \paren ... | :$\ds \int \frac {\d x} {\cos a x \paren {1 - \sin a x} } = \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {2 a} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 4} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sin a x
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = a \cos a x
| c = [[Derivative of Sine of a x|Derivative of $\sin a x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cos a x \paren {1 - \sin a x} }
| r = \int \frac {\co... | Primitive of Reciprocal of Cosine of a x by 1 minus Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cosine_of_a_x_by_1_minus_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cosine_of_a_x_by_1_minus_Sine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Derivative of Sine Function/Corollary",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sum of Squares of Sine and Cosine",
"Integration by Substitution",
"Difference of Two Squares",
"Primitive of Reciprocal of a x + b squared by p x + q",
"Reciprocal of One Minus Sine",
"Reci... |
proofwiki-9592 | Primitive of Reciprocal of Sine of a x by 1 plus Cosine of a x | :$\ds \int \frac {\d x} {\sin a x \paren {1 + \cos a x} } = \frac 1 {2 a \paren {1 + \cos a x} } + \frac 1 {2 a} \ln \size {\tan \frac {a x} 2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \cos a x
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = -a \sin a x
| c = Derivative of $\cos a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x \paren {1 + \cos a x} }
| r = \int \frac {\sin a x \rd x} {\sin^2 a x \paren... | :$\ds \int \frac {\d x} {\sin a x \paren {1 + \cos a x} } = \frac 1 {2 a \paren {1 + \cos a x} } + \frac 1 {2 a} \ln \size {\tan \frac {a x} 2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \cos a x
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = -a \sin a x
| c = [[Derivative of Cosine of a x|Derivative of $\cos a x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x \paren {1 + \cos a x} }
| r = \int \frac {... | Primitive of Reciprocal of Sine of a x by 1 plus Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x_by_1_plus_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x_by_1_plus_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Derivative of Cosine Function/Corollary",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sum of Squares of Sine and Cosine",
"Integration by Substitution",
"Difference of Two Squares",
"Primitive of Reciprocal of a x + b squared by p x + q",
"Reciprocal of One Minus Cosine",
"... |
proofwiki-9593 | Primitive of Reciprocal of Sine of a x by 1 minus Cosine of a x | :$\ds \int \frac {\d x} {\sin a x \paren {1 - \cos a x} } = \frac {-1} {2 a \paren {1 - \cos a x} } + \frac 1 {2 a} \ln \size {\tan \frac {a x} 2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \cos a x
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = -a \sin a x
| c = Derivative of $\cos a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x \paren {1 - \cos a x} }
| r = \int \frac {\sin a x \rd x} {\sin^2 a x \paren... | :$\ds \int \frac {\d x} {\sin a x \paren {1 - \cos a x} } = \frac {-1} {2 a \paren {1 - \cos a x} } + \frac 1 {2 a} \ln \size {\tan \frac {a x} 2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \cos a x
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = -a \sin a x
| c = [[Derivative of Cosine of a x|Derivative of $\cos a x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x \paren {1 - \cos a x} }
| r = \int \frac {... | Primitive of Reciprocal of Sine of a x by 1 minus Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x_by_1_minus_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x_by_1_minus_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Derivative of Cosine Function/Corollary",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sum of Squares of Sine and Cosine",
"Integration by Substitution",
"Difference of Two Squares",
"Primitive of Reciprocal of a x + b squared by p x + q",
"Logarithm of Power",
"Reciprocal o... |
proofwiki-9594 | Primitive of Reciprocal of Sine of a x plus Cosine of a x | :$\ds \int \frac {\d x} {\sin a x + \cos a x} = \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 8} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x + \cos a x}
| r = \int \frac {\d x} {\sqrt 2 \map \cos {a x - \dfrac \pi 4} }
| c = Sine of x plus Cosine of x: Cosine Form
}}
{{eqn | r = \frac 1 {\sqrt 2} \int \map \sec {a x - \dfrac \pi 4} \rd x
| c = Secant is Reciprocal of Cosine
}}
{{end-eqn... | :$\ds \int \frac {\d x} {\sin a x + \cos a x} = \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 8} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x + \cos a x}
| r = \int \frac {\d x} {\sqrt 2 \map \cos {a x - \dfrac \pi 4} }
| c = [[Sine of x plus Cosine of x/Cosine Form|Sine of x plus Cosine of x: Cosine Form]]
}}
{{eqn | r = \frac 1 {\sqrt 2} \int \map \sec {a x - \dfrac \pi 4} \rd x
| c = ... | Primitive of Reciprocal of Sine of a x plus Cosine of a x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x_plus_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x_plus_Cosine_of_a_x/Proof_1 | [
"Primitive of Reciprocal of Sine of a x plus Cosine of a x",
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Sine of x plus Cosine of x/Cosine Form",
"Secant is Reciprocal of Cosine",
"Derivative of Identity Function",
"Derivatives of Function of a x + b",
"Integration by Substitution",
"Primitive of Secant of a x/Tangent Form"
] |
proofwiki-9595 | Primitive of Reciprocal of Sine of a x plus Cosine of a x | :$\ds \int \frac {\d x} {\sin a x + \cos a x} = \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 8} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x + \cos a x}
| r = \frac 1 a \int \frac {\dfrac {2 \rd u} {1 + u^2} } {\dfrac {2 u} {1 + u^2} + \dfrac {1 - u^2} {1 + u^2} }
| c = Weierstrass Substitution: $u = \tan \dfrac {a x} 2$
}}
{{eqn | r = \frac 2 a \int \frac {\d u} {- u^2 + 2 u + 1}
| c =... | :$\ds \int \frac {\d x} {\sin a x + \cos a x} = \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 8} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x + \cos a x}
| r = \frac 1 a \int \frac {\dfrac {2 \rd u} {1 + u^2} } {\dfrac {2 u} {1 + u^2} + \dfrac {1 - u^2} {1 + u^2} }
| c = [[Weierstrass Substitution]]: $u = \tan \dfrac {a x} 2$
}}
{{eqn | r = \frac 2 a \int \frac {\d u} {- u^2 + 2 u + 1}
|... | Primitive of Reciprocal of Sine of a x plus Cosine of a x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x_plus_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x_plus_Cosine_of_a_x/Proof_2 | [
"Primitive of Reciprocal of Sine of a x plus Cosine of a x",
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Weierstrass Substitution",
"Primitive of Reciprocal of a x squared plus b x plus c",
"Tangent of 22.5 Degrees",
"Tangent of 67.5 Degrees"
] |
proofwiki-9596 | Primitive of Reciprocal of Sine of a x minus Cosine of a x | :$\ds \int \frac {\d x} {\sin a x - \cos a x} = \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 - \frac \pi 8} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x - \cos a x}
| r = \int \frac {\d x} {\sqrt 2 \map \cos {a x - \dfrac {3 \pi} 4} }
| c = Sine of x minus Cosine of x: Cosine Form
}}
{{eqn | r = \frac 1 {\sqrt 2} \int \map \sec {a x - \dfrac {3 \pi} 4} \rd x
| c = Secant is Reciprocal of Cosine
}}
... | :$\ds \int \frac {\d x} {\sin a x - \cos a x} = \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 - \frac \pi 8} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x - \cos a x}
| r = \int \frac {\d x} {\sqrt 2 \map \cos {a x - \dfrac {3 \pi} 4} }
| c = [[Sine of x minus Cosine of x/Cosine Form|Sine of x minus Cosine of x: Cosine Form]]
}}
{{eqn | r = \frac 1 {\sqrt 2} \int \map \sec {a x - \dfrac {3 \pi} 4} \rd x
... | Primitive of Reciprocal of Sine of a x minus Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x_minus_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x_minus_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Sine of x minus Cosine of x/Cosine Form",
"Secant is Reciprocal of Cosine",
"Derivative of Identity Function",
"Derivatives of Function of a x + b",
"Integration by Substitution",
"Primitive of Secant of a x/Tangent Form"
] |
proofwiki-9597 | Primitive of Sine of a x over Sine of a x plus Cosine of a x | :$\ds \int \frac {\sin a x \rd x} {\sin a x + \cos a x} = \frac x 2 - \frac 1 {2 a} \ln \size {\sin a x + \cos a x} + C$ | First note that:
{{begin-eqn}}
{{eqn | n = 1
| l = \map {\frac \d {\d x} } {\sin a x + \cos a x}
| r = a \paren {\cos a x - \sin a x}
| c = Derivative of $\sin a x$ and Derivative of $\cos a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\sin a x \rd x} {\sin a x + \cos a x}
| r =... | :$\ds \int \frac {\sin a x \rd x} {\sin a x + \cos a x} = \frac x 2 - \frac 1 {2 a} \ln \size {\sin a x + \cos a x} + C$ | First note that:
{{begin-eqn}}
{{eqn | n = 1
| l = \map {\frac \d {\d x} } {\sin a x + \cos a x}
| r = a \paren {\cos a x - \sin a x}
| c = [[Derivative of Sine of a x|Derivative of $\sin a x$]] and [[Derivative of Cosine of a x|Derivative of $\cos a x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | ... | Primitive of Sine of a x over Sine of a x plus Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_Sine_of_a_x_plus_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_Sine_of_a_x_plus_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Derivative of Sine Function/Corollary",
"Derivative of Cosine Function/Corollary",
"Linear Combination of Integrals/Indefinite",
"Linear Combination of Integrals/Indefinite",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Function under its Derivative"
] |
proofwiki-9598 | Primitive of Sine of a x over Sine of a x minus Cosine of a x | :$\ds \int \frac {\sin a x \rd x} {\sin a x - \cos a x} = \frac x 2 + \frac 1 {2 a} \ln \size {\sin a x - \cos a x} + C$ | First note that:
{{begin-eqn}}
{{eqn | n = 1
| l = \map {\frac {\d} {\d x} } {\sin a x - \cos a x}
| r = a \paren {\cos a x + \sin a x}
| c = Derivative of $\sin a x$ and Derivative of $\cos a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\sin a x \rd x} {\sin a x - \cos a x}
| r... | :$\ds \int \frac {\sin a x \rd x} {\sin a x - \cos a x} = \frac x 2 + \frac 1 {2 a} \ln \size {\sin a x - \cos a x} + C$ | First note that:
{{begin-eqn}}
{{eqn | n = 1
| l = \map {\frac {\d} {\d x} } {\sin a x - \cos a x}
| r = a \paren {\cos a x + \sin a x}
| c = [[Derivative of Sine of a x|Derivative of $\sin a x$]] and [[Derivative of Cosine of a x|Derivative of $\cos a x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn ... | Primitive of Sine of a x over Sine of a x minus Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_Sine_of_a_x_minus_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_Sine_of_a_x_minus_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Derivative of Sine Function/Corollary",
"Derivative of Cosine Function/Corollary",
"Linear Combination of Integrals/Indefinite",
"Linear Combination of Integrals/Indefinite",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Function under its Derivative"
] |
proofwiki-9599 | Primitive of Cosine of a x over Sine of a x plus 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 {\sin a x + \cos 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 {\sin a x + \cos 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 plus Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_Sine_of_a_x_plus_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_Sine_of_a_x_plus_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 plus Cosine of a x"
] |
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