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-9800 | Primitive of Reciprocal of Square of Hyperbolic Sine of a x | :$\ds \int \frac {\d x} {\sinh^2 a x} = -\frac {\coth a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh^2 a x}
| r = \int \csch^2 a x \rd x
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = -\frac {\coth a x} a + C
| c = Primitive of $\csch^2 a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {\sinh^2 a x} = -\frac {\coth a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh^2 a x}
| r = \int \csch^2 a x \rd x
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = -\frac {\coth a x} a + C
| c = [[Primitive of Square of Hyperbolic Cosecant of a x|Primitive of $\csch^2 a x$]]
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of Square of Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Primitive of Square of Hyperbolic Cosecant of a x"
] |
proofwiki-9801 | Primitive of Reciprocal of Square of Hyperbolic Cosine of a x | :$\ds \int \frac {\d x} {\cosh^2 a x} = \frac {\tanh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cosh^2 a x}
| r = \int \sech^2 a x \rd x
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac {\tanh a x} a + C
| c = Primitive of $\sech^2 a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {\cosh^2 a x} = \frac {\tanh a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cosh^2 a x}
| r = \int \sech^2 a x \rd x
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac {\tanh a x} a + C
| c = [[Primitive of Square of Hyperbolic Secant of a x|Primitive of $\sech^2 a x$]]
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of Square of Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Primitive of Square of Hyperbolic Secant of a x"
] |
proofwiki-9802 | Primitive of Hyperbolic Sine of a x by Hyperbolic Sine of p x | :$\ds \int \sinh a x \sinh p x \rd x = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } - \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh a x \sinh p x \rd x
| r = \int \paren {\frac {\map \cosh {a x + p x} - \map \cosh {a x - p x} } 2} \rd x
| c = Werner Formula for Hyperbolic Sine by Hyperbolic Sine
}}
{{eqn | r = \frac 1 2 \int \map \cosh {a + p} x \rd x - \frac 1 2 \int \map \cosh {a - p} x \rd x
... | :$\ds \int \sinh a x \sinh p x \rd x = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } - \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh a x \sinh p x \rd x
| r = \int \paren {\frac {\map \cosh {a x + p x} - \map \cosh {a x - p x} } 2} \rd x
| c = [[Werner Formula for Hyperbolic Sine by Hyperbolic Sine]]
}}
{{eqn | r = \frac 1 2 \int \map \cosh {a + p} x \rd x - \frac 1 2 \int \map \cosh {a - p} x \rd x
... | Primitive of Hyperbolic Sine of a x by Hyperbolic Sine of p x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Sine_of_p_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Sine_of_p_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Werner Formulas/Hyperbolic Sine by Hyperbolic Sine",
"Linear Combination of Integrals/Indefinite",
"Primitive of Hyperbolic Cosine of a x"
] |
proofwiki-9803 | Primitive of Hyperbolic Cosine of a x by Hyperbolic Cosine of p x | :$\ds \int \cosh a x \cosh p x \rd x = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } + \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C$ | {{begin-eqn}}
{{eqn | l = \int \cosh a x \cosh p x \rd x
| r = \int \paren {\frac {\map \cosh {a x + p x} + \map \cosh {a x - p x} } 2} \rd x
| c = Werner Formula for Hyperbolic Cosine by Hyperbolic Cosine
}}
{{eqn | r = \frac 1 2 \int \map \cosh {a + p} x \rd x + \frac 1 2 \int \map \cosh {a - p} x \rd x
... | :$\ds \int \cosh a x \cosh p x \rd x = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } + \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C$ | {{begin-eqn}}
{{eqn | l = \int \cosh a x \cosh p x \rd x
| r = \int \paren {\frac {\map \cosh {a x + p x} + \map \cosh {a x - p x} } 2} \rd x
| c = [[Werner Formula for Hyperbolic Cosine by Hyperbolic Cosine]]
}}
{{eqn | r = \frac 1 2 \int \map \cosh {a + p} x \rd x + \frac 1 2 \int \map \cosh {a - p} x \rd... | Primitive of Hyperbolic Cosine of a x by Hyperbolic Cosine of p x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosine_of_a_x_by_Hyperbolic_Cosine_of_p_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosine_of_a_x_by_Hyperbolic_Cosine_of_p_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Werner Formulas/Hyperbolic Cosine by Hyperbolic Cosine",
"Linear Combination of Integrals/Indefinite",
"Primitive of Hyperbolic Cosine of a x"
] |
proofwiki-9804 | Primitive of Hyperbolic Sine of a x by Sine of p x | :$\ds \int \sinh a x \sin p x \rd x = \frac {a \cosh a x \sin p x - p \sinh a x \cos p x} {a^2 + p^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh a x \sin p x \rd x
| r = \int \paren {\frac {e^{a x} - e^{-a x} } 2} \sin p x \rd x
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \int e^{a x} \sin p x \rd x - \frac 1 2 \int e^{- a x} \sin p x \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = ... | :$\ds \int \sinh a x \sin p x \rd x = \frac {a \cosh a x \sin p x - p \sinh a x \cos p x} {a^2 + p^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh a x \sin p x \rd x
| r = \int \paren {\frac {e^{a x} - e^{-a x} } 2} \sin p x \rd x
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \int e^{a x} \sin p x \rd x - \frac 1 2 \int e^{- a x} \sin p x \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | ... | Primitive of Hyperbolic Sine of a x by Sine of p x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Sine_of_p_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Sine_of_p_x | [
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Sine Function"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Exponential of a x by Sine of b x"
] |
proofwiki-9805 | Primitive of Hyperbolic Sine of a x by Cosine of p x | :$\ds \int \sinh a x \cos p x \rd x = \frac {a \cosh a x \cos p x + p \sinh a x \sin p x} {a^2 + p^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh a x \cos p x \rd x
| r = \int \paren {\frac {e^{a x} - e^{- a x} } 2} \cos p x \rd x
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \int e^{a x} \cos p x \rd x - \frac 1 2 \int e^{- a x} \cos p x \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r =... | :$\ds \int \sinh a x \cos p x \rd x = \frac {a \cosh a x \cos p x + p \sinh a x \sin p x} {a^2 + p^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh a x \cos p x \rd x
| r = \int \paren {\frac {e^{a x} - e^{- a x} } 2} \cos p x \rd x
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \int e^{a x} \cos p x \rd x - \frac 1 2 \int e^{- a x} \cos p x \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn |... | Primitive of Hyperbolic Sine of a x by Cosine of p x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Cosine_of_p_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Cosine_of_p_x | [
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Cosine Function"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Exponential of a x by Cosine of b x"
] |
proofwiki-9806 | Primitive of Hyperbolic Cosine of a x by Sine of p x | :$\ds \int \cosh a x \sin p x \rd x = \frac {a \sinh a x \sin p x - p \cosh a x \cos p x} {a^2 + p^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \cosh a x \sin p x \rd x
| r = \int \paren {\frac {e^{a x} + e^{-a x} } 2} \sin p x \rd x
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac 1 2 \int e^{a x} \sin p x \rd x + \frac 1 2 \int e^{- a x} \sin p x \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r ... | :$\ds \int \cosh a x \sin p x \rd x = \frac {a \sinh a x \sin p x - p \cosh a x \cos p x} {a^2 + p^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \cosh a x \sin p x \rd x
| r = \int \paren {\frac {e^{a x} + e^{-a x} } 2} \sin p x \rd x
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac 1 2 \int e^{a x} \sin p x \rd x + \frac 1 2 \int e^{- a x} \sin p x \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn ... | Primitive of Hyperbolic Cosine of a x by Sine of p x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosine_of_a_x_by_Sine_of_p_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosine_of_a_x_by_Sine_of_p_x | [
"Primitives involving Hyperbolic Cosine Function",
"Primitives involving Sine Function"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Exponential of a x by Sine of b x"
] |
proofwiki-9807 | Primitive of Hyperbolic Cosine of a x by Cosine of p x | :$\ds \int \cosh a x \cos p x \rd x = \frac {a \sinh a x \cos p x + p \cosh a x \sin p x} {a^2 + p^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \cosh a x \cos p x \rd x
| r = \int \paren {\frac {e^{a x} + e^{- a x} } 2} \cos p x \rd x
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac 1 2 \int e^{a x} \cos p x \rd x + \frac 1 2 \int e^{- a x} \cos p x \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r... | :$\ds \int \cosh a x \cos p x \rd x = \frac {a \sinh a x \cos p x + p \cosh a x \sin p x} {a^2 + p^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \cosh a x \cos p x \rd x
| r = \int \paren {\frac {e^{a x} + e^{- a x} } 2} \cos p x \rd x
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac 1 2 \int e^{a x} \cos p x \rd x + \frac 1 2 \int e^{- a x} \cos p x \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn... | Primitive of Hyperbolic Cosine of a x by Cosine of p x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosine_of_a_x_by_Cosine_of_p_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosine_of_a_x_by_Cosine_of_p_x | [
"Primitives involving Hyperbolic Cosine Function",
"Primitives involving Cosine Function"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Exponential of a x by Cosine of b x"
] |
proofwiki-9808 | Primitive of Reciprocal of p plus q by Hyperbolic Sine of a x | :$\ds \int \frac {\d x} {p + q \sinh a x} = \frac 1 {a \sqrt{p^2 + q^2} } \ln \size {\frac {q e^{a x} + p - \sqrt {p^2 + q^2} } {q e^{a x} + p + \sqrt {p^2 + q^2} } } + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = e^{a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d u} {\d x}
| r = e^{a x} = u
| c =
}}
{{eqn | ll= \leadsto
| l = \d x
| r = \dfrac {\d u} u
| c =
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \sinh... | :$\ds \int \frac {\d x} {p + q \sinh a x} = \frac 1 {a \sqrt{p^2 + q^2} } \ln \size {\frac {q e^{a x} + p - \sqrt {p^2 + q^2} } {q e^{a x} + p + \sqrt {p^2 + q^2} } } + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = e^{a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d u} {\d x}
| r = e^{a x} = u
| c =
}}
{{eqn | ll= \leadsto
| l = \d x
| r = \dfrac {\d u} u
| c =
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \... | Primitive of Reciprocal of p plus q by Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Integration by Substitution",
"Definition:Discriminant",
"Primitive of Reciprocal of a x squared plus b x plus c"
] |
proofwiki-9809 | Primitive of Reciprocal of p plus q by Hyperbolic Cosine of a x | For $p \ne q$:
:$\ds \int \frac {\d x} {p + q \cosh a x} = \begin {cases}
\dfrac 2 {a \sqrt {q^2 - p^2} } \arctan \dfrac {q e^{a x} + p} {\sqrt {q^2 - p^2} } + C & : p^2 < q^2 \\
\dfrac 1 {a \sqrt {p^2 - q^2} } \ln \size {\dfrac {q e^{a x} + p - \sqrt {p^2 - q^2} } {q e^{a x} + p + \sqrt {p^2 - q^2} } } + C & : p^2 > q... | Let:
{{begin-eqn}}
{{eqn | l = u
| r = e^{a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d u} {\d x}
| r = e^{a x} = u
| c =
}}
{{eqn | ll= \leadsto
| l = \d x
| r = \dfrac {\d u} u
| c =
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \cosh... | For $p \ne q$:
:$\ds \int \frac {\d x} {p + q \cosh a x} = \begin {cases}
\dfrac 2 {a \sqrt {q^2 - p^2} } \arctan \dfrac {q e^{a x} + p} {\sqrt {q^2 - p^2} } + C & : p^2 < q^2 \\
\dfrac 1 {a \sqrt {p^2 - q^2} } \ln \size {\dfrac {q e^{a x} + p - \sqrt {p^2 - q^2} } {q e^{a x} + p + \sqrt {p^2 - q^2} } } + C & : p^2 > ... | Let:
{{begin-eqn}}
{{eqn | l = u
| r = e^{a x}
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d u} {\d x}
| r = e^{a x} = u
| c =
}}
{{eqn | ll= \leadsto
| l = \d x
| r = \dfrac {\d u} u
| c =
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \... | Primitive of Reciprocal of p plus q by Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Integration by Substitution",
"Definition:Discriminant",
"Primitive of Reciprocal of a x squared plus b x plus c",
"Primitive of Reciprocal of a x squared plus b x plus c"
] |
proofwiki-9810 | Primitive of Reciprocal of p plus q by Hyperbolic Tangent of a x | :$\ds \int \frac {\d x} {p + q \tanh a x} = \frac {p x} {p^2 - q^2} - \frac q {a \paren {p^2 - q^2} } \ln \size {q \sinh a x + p \cosh a x} + C$ | We have:
:$\dfrac \d {\d x} \paren {q \sinh a x + p \cosh a x} = a q \cosh a x + a p \sinh a x$
Thus:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \tanh a x}
| r = \int \frac {\d x} {p + q \dfrac {\sinh a x} {\cosh a x} }
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \int \frac {\cosh a ... | :$\ds \int \frac {\d x} {p + q \tanh a x} = \frac {p x} {p^2 - q^2} - \frac q {a \paren {p^2 - q^2} } \ln \size {q \sinh a x + p \cosh a x} + C$ | We have:
:$\dfrac \d {\d x} \paren {q \sinh a x + p \cosh a x} = a q \cosh a x + a p \sinh a x$
Thus:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \tanh a x}
| r = \int \frac {\d x} {p + q \dfrac {\sinh a x} {\cosh a x} }
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \int \frac {\cosh ... | Primitive of Reciprocal of p plus q by Hyperbolic Tangent of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Hyperbolic_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Hyperbolic_Tangent_of_a_x | [
"Primitives involving Hyperbolic Tangent Function"
] | [] | [
"Primitive of Constant",
"Primitive of Reciprocal"
] |
proofwiki-9811 | Primitive of Reciprocal of p plus q by Hyperbolic Cotangent of a x | :$\ds \int \frac {\d x} {p + q \coth a x} = \frac {p x} {p^2 - q^2} - \frac q {a \paren {p^2 - q^2} } \ln \size {p \sinh a x + q \cosh a x} + C$ | We have:
:$\dfrac \d {\d x} \paren {p \sinh a x + q \cosh a x} = a p \cosh a x + a q \sinh a x$
Thus:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \coth a x}
| r = \int \frac {\d x} {p + q \dfrac {\cosh a x} {\sinh a x} }
| c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \int \frac {\sinh ... | :$\ds \int \frac {\d x} {p + q \coth a x} = \frac {p x} {p^2 - q^2} - \frac q {a \paren {p^2 - q^2} } \ln \size {p \sinh a x + q \cosh a x} + C$ | We have:
:$\dfrac \d {\d x} \paren {p \sinh a x + q \cosh a x} = a p \cosh a x + a q \sinh a x$
Thus:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \coth a x}
| r = \int \frac {\d x} {p + q \dfrac {\cosh a x} {\sinh a x} }
| c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \int \frac {\sin... | Primitive of Reciprocal of p plus q by Hyperbolic Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Hyperbolic_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Hyperbolic_Cotangent_of_a_x | [
"Primitives involving Hyperbolic Cotangent Function"
] | [] | [
"Primitive of Constant",
"Primitive of Reciprocal"
] |
proofwiki-9812 | Primitive of Reciprocal of q plus p by Hyperbolic Secant of a x | :$\ds \int \frac {\d x} {q + p \sech a x} = \frac x q - \frac p q \int \frac {\d x} {p + q \cosh a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {q + p \sech a x}
| r = \frac 1 q \int \frac {q \rd x} {q + p \sech a x}
| c = multiplying top and bottom by $q$
}}
{{eqn | r = \frac 1 q \int \frac {\paren {q + p \sech a x - p \sech a x} \rd x} {q + p \sech a x}
| c =
}}
{{eqn | r = \frac 1 q \int \frac ... | :$\ds \int \frac {\d x} {q + p \sech a x} = \frac x q - \frac p q \int \frac {\d x} {p + q \cosh a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {q + p \sech a x}
| r = \frac 1 q \int \frac {q \rd x} {q + p \sech a x}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $q$
}}
{{eqn | r = \frac 1 q \int \frac {\paren {q + p \sech a x - p \sech a x} \rd x} {q + p \sech a ... | Primitive of Reciprocal of q plus p by Hyperbolic Secant of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_q_plus_p_by_Hyperbolic_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_q_plus_p_by_Hyperbolic_Secant_of_a_x | [
"Primitives involving Hyperbolic Secant Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-9813 | Primitive of Reciprocal of q plus p by Hyperbolic Cosecant of a x | :$\ds \int \frac {\d x} {q + p \csch a x} = \frac x q - \frac p q \int \frac {\d x} {p + q \sinh a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {q + p \csch a x}
| r = \frac 1 q \int \frac {q \rd x} {q + p \csch a x}
| c = multiplying top and bottom by $q$
}}
{{eqn | r = \frac 1 q \int \frac {\paren {q + p \csch a x - p \csch a x} \rd x} {q + p \csch a x}
| c =
}}
{{eqn | r = \frac 1 q \int \frac {... | :$\ds \int \frac {\d x} {q + p \csch a x} = \frac x q - \frac p q \int \frac {\d x} {p + q \sinh a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {q + p \csch a x}
| r = \frac 1 q \int \frac {q \rd x} {q + p \csch a x}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $q$
}}
{{eqn | r = \frac 1 q \int \frac {\paren {q + p \csch a x - p \csch a x} \rd x} {q + p \csch a x... | Primitive of Reciprocal of q plus p by Hyperbolic Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_q_plus_p_by_Hyperbolic_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_q_plus_p_by_Hyperbolic_Cosecant_of_a_x | [
"Primitives involving Hyperbolic Cosecant Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-9814 | Primitive of Reciprocal of Square of p plus q by Hyperbolic Sine of a x | :$\ds \int \frac {\d x} {\paren {p + q \sinh a x}^2} = \frac {-q \cosh a x} {a \paren {p^2 + q^2} \paren {p + q \sinh a x} } + \frac p {p^2 + q^2} \int \frac {\d x} {p + q \sinh a x} + C$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\dfrac {\cosh a x} {p + q \sinh a x} }
| r = \dfrac {\paren {p + q \sinh a x} \map {\frac \d {\d x} } {\cosh a x} - \cosh a x \map {\frac \d {\d x} } {p + q \sinh a x} } {\paren {p + q \sinh a x}^2}
| c = Quotient Rule for Derivatives
}}
{{eqn | r = \dfrac... | :$\ds \int \frac {\d x} {\paren {p + q \sinh a x}^2} = \frac {-q \cosh a x} {a \paren {p^2 + q^2} \paren {p + q \sinh a x} } + \frac p {p^2 + q^2} \int \frac {\d x} {p + q \sinh a x} + C$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\dfrac {\cosh a x} {p + q \sinh a x} }
| r = \dfrac {\paren {p + q \sinh a x} \map {\frac \d {\d x} } {\cosh a x} - \cosh a x \map {\frac \d {\d x} } {p + q \sinh a x} } {\paren {p + q \sinh a x}^2}
| c = [[Quotient Rule for Derivatives]]
}}
{{eqn | r = \d... | Primitive of Reciprocal of Square of p plus q by Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_p_plus_q_by_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_p_plus_q_by_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Quotient Rule for Derivatives",
"Derivative of Hyperbolic Cosine Function",
"Derivative of Hyperbolic Sine Function",
"Difference of Squares of Hyperbolic Cosine and Sine"
] |
proofwiki-9815 | Primitive of Reciprocal of Square of p plus q by Hyperbolic Cosine of a x | :$\ds \int \frac {\d x} {\paren {p + q \cosh a x}^2} = \frac {q \sinh a x} {a \paren {q^2 - p^2} \paren {p + q \cosh a x} } - \frac p {q^2 - p^2} \int \frac {\rd x} {p + q \cosh a x} + C$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\dfrac {\sinh a x} {p + q \cosh a x} }
| r = \dfrac {\paren {p + q \cosh a x} \map {\frac \d {\d x} } {\sinh a x} - \sinh a x \map {\frac \d {\d x} } {p + q \sinh a x} } {\paren {p + q \cosh a x}^2}
| c = Quotient Rule for Derivatives
}}
{{eqn | r = \dfrac... | :$\ds \int \frac {\d x} {\paren {p + q \cosh a x}^2} = \frac {q \sinh a x} {a \paren {q^2 - p^2} \paren {p + q \cosh a x} } - \frac p {q^2 - p^2} \int \frac {\rd x} {p + q \cosh a x} + C$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\dfrac {\sinh a x} {p + q \cosh a x} }
| r = \dfrac {\paren {p + q \cosh a x} \map {\frac \d {\d x} } {\sinh a x} - \sinh a x \map {\frac \d {\d x} } {p + q \sinh a x} } {\paren {p + q \cosh a x}^2}
| c = [[Quotient Rule for Derivatives]]
}}
{{eqn | r = \d... | Primitive of Reciprocal of Square of p plus q by Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_p_plus_q_by_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_p_plus_q_by_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Quotient Rule for Derivatives",
"Derivative of Hyperbolic Cosine Function",
"Derivative of Hyperbolic Sine Function",
"Difference of Squares of Hyperbolic Cosine and Sine"
] |
proofwiki-9816 | Primitive of Reciprocal of p squared plus Square of q by Hyperbolic Sine of a x | :$\ds \int \frac {\d x} {p^2 + q^2 \sinh^2 a x} = \begin {cases}
\dfrac 1 {a p \sqrt{q^2 - p^2} } \arctan \dfrac {\sqrt {q^2 - p^2} \tanh a x} p + C & : p^2 < q^2 \\
\dfrac 1 {2 a p \sqrt{p^2 - q^2} } \ln \size {\dfrac {p + \sqrt {p^2 - q^2} \tanh a x} {p - \sqrt {p^2 - q^2} \tanh a x} } + C & : p^2 > q^2 \\
\end {case... | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 + q^2 \sinh^2 a x}
| r = \int \frac {\sech^2 a x \rd x} {p^2 \sech^2 a x + q^2 \tanh^2 a x}
| c = multiplying numerator and denominator by $\sech^2 a x$
}}
{{eqn | r = \int \frac {\sech^2 a x \rd x} {p^2 \paren {1 - \tanh^2 a x} + q^2 \tanh^2 a x}
| c =... | :$\ds \int \frac {\d x} {p^2 + q^2 \sinh^2 a x} = \begin {cases}
\dfrac 1 {a p \sqrt{q^2 - p^2} } \arctan \dfrac {\sqrt {q^2 - p^2} \tanh a x} p + C & : p^2 < q^2 \\
\dfrac 1 {2 a p \sqrt{p^2 - q^2} } \ln \size {\dfrac {p + \sqrt {p^2 - q^2} \tanh a x} {p - \sqrt {p^2 - q^2} \tanh a x} } + C & : p^2 > q^2 \\
\end {case... | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 + q^2 \sinh^2 a x}
| r = \int \frac {\sech^2 a x \rd x} {p^2 \sech^2 a x + q^2 \tanh^2 a x}
| c = multiplying [[Definition:Numerator|numerator]] and [[Definition:Denominator|denominator]] by $\sech^2 a x$
}}
{{eqn | r = \int \frac {\sech^2 a x \rd x} {p^2 \pa... | Primitive of Reciprocal of p squared plus Square of q by Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_plus_Square_of_q_by_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_plus_Square_of_q_by_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sum of Squares of Hyperbolic Secant and Tangent",
"Derivative of Hyperbolic Tangent Function",
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Primitive of Reciprocal of x squared ... |
proofwiki-9817 | Primitive of Reciprocal of p squared plus Square of q by Hyperbolic Cosine of a x/Logarithm Form | :$\ds \int \frac {\d x} {p^2 + q^2 \cosh^2 a x} = \dfrac 1 {2 a p \sqrt {p^2 + q^2} } \ln \size {\dfrac {p \tanh a x + \sqrt {p^2 + q^2} } {p \tanh a x - \sqrt {p^2 + q^2} } }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 + q^2 \cosh^2 a x}
| r = \int \frac {\csch^2 a x \rd x} {p^2 \csch^2 a x + q^2 \coth^2 a x}
| c = multiplying numerator and denominator by $\csch^2 a x$
}}
{{eqn | r = \int \frac {\csch^2 a x \rd x} {p^2 \paren {\coth^2 a x - 1} + q^2 \coth^2 a x}
| c =... | :$\ds \int \frac {\d x} {p^2 + q^2 \cosh^2 a x} = \dfrac 1 {2 a p \sqrt {p^2 + q^2} } \ln \size {\dfrac {p \tanh a x + \sqrt {p^2 + q^2} } {p \tanh a x - \sqrt {p^2 + q^2} } }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 + q^2 \cosh^2 a x}
| r = \int \frac {\csch^2 a x \rd x} {p^2 \csch^2 a x + q^2 \coth^2 a x}
| c = multiplying [[Definition:Numerator|numerator]] and [[Definition:Denominator|denominator]] by $\csch^2 a x$
}}
{{eqn | r = \int \frac {\csch^2 a x \rd x} {p^2 \pa... | Primitive of Reciprocal of p squared plus Square of q by Hyperbolic Cosine of a x/Logarithm Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_plus_Square_of_q_by_Hyperbolic_Cosine_of_a_x/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_plus_Square_of_q_by_Hyperbolic_Cosine_of_a_x/Logarithm_Form | [
"Primitive of Reciprocal of p squared plus Square of q by Hyperbolic Cosine of a x"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Derivative of Hyperbolic Cotangent Function",
"Integration by Substitution",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2",
"Definition:Fraction/Nu... |
proofwiki-9818 | Primitive of Reciprocal of p squared minus Square of q by Hyperbolic Sine of a x | :$\ds \int \frac {\d x} {p^2 - q^2 \sinh^2 a x} = \frac 1 {2 a p \sqrt {p^2 + q^2} } \ln \size {\frac {p + \sqrt {p^2 + q^2} \tanh a x} {p - \sqrt {p^2 + q^2} \tanh a x} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 - q^2 \sinh^2 a x}
| r = \int \frac {\sech^2 a x \rd x} {p^2 \sech^2 a x - q^2 \tanh^2 a x}
| c = multiplying numerator and denominator by $\sech^2 a x$
}}
{{eqn | r = \int \frac {\sech^2 a x \rd x} {p^2 \paren {1 - \tanh^2 a x} - q^2 \tanh^2 a x}
| c =... | :$\ds \int \frac {\d x} {p^2 - q^2 \sinh^2 a x} = \frac 1 {2 a p \sqrt {p^2 + q^2} } \ln \size {\frac {p + \sqrt {p^2 + q^2} \tanh a x} {p - \sqrt {p^2 + q^2} \tanh a x} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 - q^2 \sinh^2 a x}
| r = \int \frac {\sech^2 a x \rd x} {p^2 \sech^2 a x - q^2 \tanh^2 a x}
| c = multiplying [[Definition:Numerator|numerator]] and [[Definition:Denominator|denominator]] by $\sech^2 a x$
}}
{{eqn | r = \int \frac {\sech^2 a x \rd x} {p^2 \pa... | Primitive of Reciprocal of p squared minus Square of q by Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_minus_Square_of_q_by_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_minus_Square_of_q_by_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sum of Squares of Hyperbolic Secant and Tangent",
"Derivative of Hyperbolic Tangent Function",
"Integration by Substitution",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2",
"Logarithm of Reciprocal"
] |
proofwiki-9819 | Primitive of Reciprocal of p squared minus Square of q by Hyperbolic Cosine of a x | :$\ds \int \frac {\d x} {p^2 - q^2 \cosh^2 a x} = \begin {cases}
\dfrac 1 {2 a p \sqrt {p^2 - q^2} } \ln \size {\dfrac {p \tanh a x + \sqrt {p^2 - q^2} } {p \tanh a x - \sqrt {p^2 - q^2} } } + C & : p^2 > q^2 \\
\dfrac 1 {a p \sqrt {q^2 - p^2} } \arctan \dfrac {p \tanh a x} {\sqrt {q^2 - p^2} } + C & : p^2 < q^2 \\
\en... | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 - q^2 \cosh^2 a x}
| r = \int \frac {\csch^2 a x \rd x} {p^2 \csch^2 a x - q^2 \coth^2 a x}
| c = multiplying numerator and denominator by $\csch^2 a x$
}}
{{eqn | r = \int \frac {\csch^2 a x \rd x} {p^2 \paren {\coth^2 a x - 1} - q^2 \coth^2 a x}
| c =... | :$\ds \int \frac {\d x} {p^2 - q^2 \cosh^2 a x} = \begin {cases}
\dfrac 1 {2 a p \sqrt {p^2 - q^2} } \ln \size {\dfrac {p \tanh a x + \sqrt {p^2 - q^2} } {p \tanh a x - \sqrt {p^2 - q^2} } } + C & : p^2 > q^2 \\
\dfrac 1 {a p \sqrt {q^2 - p^2} } \arctan \dfrac {p \tanh a x} {\sqrt {q^2 - p^2} } + C & : p^2 < q^2 \\
\en... | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 - q^2 \cosh^2 a x}
| r = \int \frac {\csch^2 a x \rd x} {p^2 \csch^2 a x - q^2 \coth^2 a x}
| c = multiplying [[Definition:Numerator|numerator]] and [[Definition:Denominator|denominator]] by $\csch^2 a x$
}}
{{eqn | r = \int \frac {\csch^2 a x \rd x} {p^2 \pa... | Primitive of Reciprocal of p squared minus Square of q by Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_minus_Square_of_q_by_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_minus_Square_of_q_by_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Derivative of Hyperbolic Cotangent Function",
"Integration by Substitution",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2",
"Definition:Fraction/Nu... |
proofwiki-9820 | Primitive of Power of x by Hyperbolic Sine of a x | :$\ds \int x^m \sinh a x \rd x = \frac {x^m \cosh a x} a - \frac m a \int x^{m - 1} \cosh a x \rd x$ | 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 = x^m
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = m x^{m - 1}
| c = Derivative of Power
}}
{{end-eqn... | :$\ds \int x^m \sinh a x \rd x = \frac {x^m \cosh a x} a - \frac m a \int x^{m - 1} \cosh a x \rd x$ | 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 = x^m
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = m x^{m - 1}
| c = [[Deri... | Primitive of Power of x by Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Primitive of Sine Function/Corollary",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9821 | Primitive of Power of x by Hyperbolic Cosine of a x | :$\ds \int x^m \cosh a x \rd x = \frac {x^m \sinh a x} a - \frac m a \int x^{m - 1} \sinh a x \rd x + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x^m
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = m x^{m - 1}
| c = Derivative of Power
}}
{{end-eqn... | :$\ds \int x^m \cosh a x \rd x = \frac {x^m \sinh a x} a - \frac m a \int x^{m - 1} \sinh a x \rd x + C$ | With a view to expressing the [[Definition:Primitive (Calculus)|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x^m
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = m x^{m - 1}
|... | Primitive of Power of x by Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Definition:Primitive (Calculus)",
"Power Rule for Derivatives",
"Primitive of Cosine Function/Corollary",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9822 | Primitive of Power of Hyperbolic Sine of a x | :$\ds \int \sinh^n a x \rd x = \frac {\sinh^{n - 1} a x \cosh a x} {a n} - \frac {n - 1} n \int \sinh^{n - 2} a x \rd x$
for $n \ne 0$. | With a view to expressing the problem 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 = \sinh^{n - 1} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \paren {n - 1} a \sinh^{n - 2} a x \cosh a x... | :$\ds \int \sinh^n a x \rd x = \frac {\sinh^{n - 1} a x \cosh a x} {a n} - \frac {n - 1} n \int \sinh^{n - 2} a x \rd x$
for $n \ne 0$. | With a view to expressing the problem 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 = \sinh^{n - 1} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \paren {n - 1} a \sinh^{n - 2} a x \cosh a ... | Primitive of Power of Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Hyperbolic Sine of a x",
"Power Rule for Derivatives",
"Primitive of Hyperbolic Sine of a x",
"Integration by Parts",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9823 | Primitive of Power of Hyperbolic Cosine of a x | :$\ds \int \cosh^n a x \rd x = \frac {\cosh^{n - 1} a x \sinh a x} {a n} + \frac {n - 1} n \int \cosh^{n - 2} a x \rd x$
for $n \ne 0$. | With a view to expressing the problem 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 = \cosh^{n - 1} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \paren {n - 1} a \cosh^{n - 2} a x \sinh a x... | :$\ds \int \cosh^n a x \rd x = \frac {\cosh^{n - 1} a x \sinh a x} {a n} + \frac {n - 1} n \int \cosh^{n - 2} a x \rd x$
for $n \ne 0$. | With a view to expressing the problem 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 = \cosh^{n - 1} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \paren {n - 1} a \cosh^{n - 2} a x \sinh a ... | Primitive of Power of Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Hyperbolic Cosine of a x",
"Power Rule for Derivatives",
"Primitive of Hyperbolic Cosine of a x",
"Integration by Parts",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9824 | Primitive of Power of Hyperbolic Tangent of a x | :$\ds \int \tanh^n a x \rd x = \frac {-\tanh^{n - 1} a x} {a \paren {n - 1} } + \int \tanh^{n - 2} a x \rd x + C$ | {{begin-eqn}}
{{eqn | l = \int \tanh^n a x \rd x
| r = \int \tanh^{n - 2} a x \tanh^2 a x \rd x
| c =
}}
{{eqn | r = \int \tanh^{n - 2} a x \paren {1 - \sech^2 a x} \rd x
| c = Sum of Squares of Hyperbolic Secant and Tangent
}}
{{eqn | r = -\int \tanh^{n - 2} a x \sech^2 a x \rd x + \int \tanh^{n - 2... | :$\ds \int \tanh^n a x \rd x = \frac {-\tanh^{n - 1} a x} {a \paren {n - 1} } + \int \tanh^{n - 2} a x \rd x + C$ | {{begin-eqn}}
{{eqn | l = \int \tanh^n a x \rd x
| r = \int \tanh^{n - 2} a x \tanh^2 a x \rd x
| c =
}}
{{eqn | r = \int \tanh^{n - 2} a x \paren {1 - \sech^2 a x} \rd x
| c = [[Sum of Squares of Hyperbolic Secant and Tangent]]
}}
{{eqn | r = -\int \tanh^{n - 2} a x \sech^2 a x \rd x + \int \tanh^{n... | Primitive of Power of Hyperbolic Tangent of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Tangent_of_a_x | [
"Primitives involving Hyperbolic Tangent Function"
] | [] | [
"Sum of Squares of Hyperbolic Secant and Tangent",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power of Hyperbolic Tangent of a x by Square of Hyperbolic Secant of a x"
] |
proofwiki-9825 | Primitive of Power of Hyperbolic Tangent of a x by Square of Hyperbolic Secant of a x | :$\ds \int \tanh^n a x \sech^2 a x \rd x = \frac {\tanh^{n + 1} a x} {\paren {n + 1} a} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \tanh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a \sech^2 a x
| c = Derivative of $\tanh a x$
}}
{{eqn | ll= \leadsto
| l = \int \tanh^n a x \sech^2 a x \rd x
| r = \int \frac 1 a z^n \rd z
| c = Integration by Su... | :$\ds \int \tanh^n a x \sech^2 a x \rd x = \frac {\tanh^{n + 1} a x} {\paren {n + 1} a} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \tanh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a \sech^2 a x
| c = [[Derivative of Hyperbolic Tangent of a x|Derivative of $\tanh a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \tanh^n a x \sech^2 a x \rd x
| r = \int \frac... | Primitive of Power of Hyperbolic Tangent of a x by Square of Hyperbolic Secant of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Tangent_of_a_x_by_Square_of_Hyperbolic_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Tangent_of_a_x_by_Square_of_Hyperbolic_Secant_of_a_x | [
"Primitives involving Hyperbolic Tangent Function",
"Primitives involving Hyperbolic Secant Function"
] | [] | [
"Derivative of Hyperbolic Tangent of a x",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9826 | Derivative of Hyperbolic Tangent of a x | :$\map {\dfrac \d {\d x} } {\tanh a x} = a \sech^2 a x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\tanh x}
| r = \sech^2 x
| c = Derivative of $\tanh x$
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\tanh a x}
| r = a \sech^2 a x
| c = Derivative of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\tanh a x} = a \sech^2 a x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\tanh x}
| r = \sech^2 x
| c = [[Derivative of Hyperbolic Tangent|Derivative of $\tanh x$]]
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\tanh a x}
| r = a \sech^2 a x
| c = [[Derivative of Function of Constant Multiple]]
}}
{{... | Derivative of Hyperbolic Tangent of a x | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Tangent_of_a_x | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Tangent_of_a_x | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Tangent Function"
] | [] | [
"Derivative of Hyperbolic Tangent",
"Derivative of Function of Constant Multiple"
] |
proofwiki-9827 | Derivative of Hyperbolic Cotangent of a x | :$\map {\dfrac \d {\d x} } {\coth a x} = -a \csch^2 a x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\coth x}
| r = -\csch^2 x
| c = Derivative of $\coth x$
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\coth a x}
| r = -a \csch^2 a x
| c = Derivative of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\coth a x} = -a \csch^2 a x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\coth x}
| r = -\csch^2 x
| c = [[Derivative of Hyperbolic Cotangent|Derivative of $\coth x$]]
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\coth a x}
| r = -a \csch^2 a x
| c = [[Derivative of Function of Constant Multiple]]
}... | Derivative of Hyperbolic Cotangent of a x | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cotangent_of_a_x | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cotangent_of_a_x | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Cotangent Function"
] | [] | [
"Derivative of Hyperbolic Cotangent",
"Derivative of Function of Constant Multiple"
] |
proofwiki-9828 | Primitive of Power of Hyperbolic Cotangent of a x by Square of Hyperbolic Cosecant of a x | :$\ds \int \coth^n a x \csch^2 a x \rd x = \frac {-\coth^{n + 1} a x} {\paren {n + 1} a} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \coth a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -a \csch^2 a x
| c = Derivative of $\coth a x$
}}
{{eqn | ll= \leadsto
| l = \int \coth^n a x \csch^2 a x \rd x
| r = \int \frac {-1} a z^n \rd z
| c = Integration b... | :$\ds \int \coth^n a x \csch^2 a x \rd x = \frac {-\coth^{n + 1} a x} {\paren {n + 1} a} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \coth a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -a \csch^2 a x
| c = [[Derivative of Hyperbolic Cotangent of a x|Derivative of $\coth a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \coth^n a x \csch^2 a x \rd x
| r = \int \f... | Primitive of Power of Hyperbolic Cotangent of a x by Square of Hyperbolic Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Cotangent_of_a_x_by_Square_of_Hyperbolic_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Cotangent_of_a_x_by_Square_of_Hyperbolic_Cosecant_of_a_x | [
"Primitives involving Hyperbolic Cotangent Function",
"Primitives involving Hyperbolic Cosecant Function"
] | [] | [
"Derivative of Hyperbolic Cotangent of a x",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9829 | Primitive of Power of Hyperbolic Cotangent of a x | :$\ds \int \coth^n a x \rd x = \frac {-\coth^{n - 1} a x} {a \paren {n - 1} } + \int \coth^{n - 2} a x \rd x + C$ | {{begin-eqn}}
{{eqn | l = \int \coth^n a x \rd x
| r = \int \coth^{n - 2} a x \coth^2 a x \rd x
| c =
}}
{{eqn | r = \int \coth^{n - 2} a x \paren {1 + \csch^2 a x} \rd x
| c = Difference of Squares of Hyperbolic Cotangent and Cosecant
}}
{{eqn | r = \int \coth^{n - 2} a x \csch^2 a x \rd x + \int \c... | :$\ds \int \coth^n a x \rd x = \frac {-\coth^{n - 1} a x} {a \paren {n - 1} } + \int \coth^{n - 2} a x \rd x + C$ | {{begin-eqn}}
{{eqn | l = \int \coth^n a x \rd x
| r = \int \coth^{n - 2} a x \coth^2 a x \rd x
| c =
}}
{{eqn | r = \int \coth^{n - 2} a x \paren {1 + \csch^2 a x} \rd x
| c = [[Difference of Squares of Hyperbolic Cotangent and Cosecant]]
}}
{{eqn | r = \int \coth^{n - 2} a x \csch^2 a x \rd x + \in... | Primitive of Power of Hyperbolic Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Cotangent_of_a_x | [
"Primitives involving Hyperbolic Cotangent Function"
] | [] | [
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power of Hyperbolic Cotangent of a x by Square of Hyperbolic Cosecant of a x"
] |
proofwiki-9830 | Derivative of Hyperbolic Secant of a x | :$\map {\dfrac \d {\d x} } {\sech a x} = -a \sech a x \tanh a x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\sech x}
| r = -\sech x \tanh x
| c = Derivative of $\sech x$
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\sech a x}
| r = -a \sech a x \tanh a x
| c = Derivative of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\sech a x} = -a \sech a x \tanh a x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\sech x}
| r = -\sech x \tanh x
| c = [[Derivative of Hyperbolic Secant|Derivative of $\sech x$]]
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\sech a x}
| r = -a \sech a x \tanh a x
| c = [[Derivative of Function of Constant M... | Derivative of Hyperbolic Secant of a x | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Secant_of_a_x | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Secant_of_a_x | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Secant Function"
] | [] | [
"Derivative of Hyperbolic Secant",
"Derivative of Function of Constant Multiple"
] |
proofwiki-9831 | Derivative of Hyperbolic Cosecant of a x | :$\map {\dfrac \d {\d x} } {\csch a x} = -a \csch a x \coth a x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\csch x}
| r = -\csch x \coth x
| c = Derivative of $\csch x$
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\csch a x}
| r = -a \csch a x \coth a x
| c = Derivative of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\csch a x} = -a \csch a x \coth a x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\csch x}
| r = -\csch x \coth x
| c = [[Derivative of Hyperbolic Cosecant|Derivative of $\csch x$]]
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\csch a x}
| r = -a \csch a x \coth a x
| c = [[Derivative of Function of Constant... | Derivative of Hyperbolic Cosecant of a x | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cosecant_of_a_x | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cosecant_of_a_x | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Cosecant Function"
] | [] | [
"Derivative of Hyperbolic Cosecant",
"Derivative of Function of Constant Multiple"
] |
proofwiki-9832 | Primitive of Power of Hyperbolic Secant of a x | :$\ds \int \sech^n a x \rd x = \frac {\sech^{n - 2} a x \tanh a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sech^{n - 2} a x \rd x + C$
for $n \ne -1$. | 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 = \sech^{n - 2} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -a \paren {n - 2} \sech^{n - 3} a x \sech ... | :$\ds \int \sech^n a x \rd x = \frac {\sech^{n - 2} a x \tanh a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sech^{n - 2} a x \rd x + C$
for $n \ne -1$. | 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 = \sech^{n - 2} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -a \paren {n - 2... | Primitive of Power of Hyperbolic Secant of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Secant_of_a_x | [
"Primitives involving Hyperbolic Secant Function"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Derivative of Hyperbolic Secant of a x",
"Derivative of Composite Function",
"Primitive of Square of Hyperbolic Secant of a x",
"Integration by Parts",
"Sum of Squares of Hyperbolic Secant and Tangent",
"Linear Combination of Integrals/Indefinite"... |
proofwiki-9833 | Primitive of Power of Hyperbolic Cosecant of a x | :$\ds \int \csch^n a x \rd x = \frac {-\csch^{n - 2} a x \coth a x} {a \paren {n - 1} } - \frac {n - 2} {n - 1} \int \csch^{n - 2} a x \rd x + C$
for $n \ne -1$. | 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 = \csch^{n - 2} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -a \paren {n - 2} \csch^{n - 3} a x \csch ... | :$\ds \int \csch^n a x \rd x = \frac {-\csch^{n - 2} a x \coth a x} {a \paren {n - 1} } - \frac {n - 2} {n - 1} \int \csch^{n - 2} a x \rd x + C$
for $n \ne -1$. | 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 = \csch^{n - 2} a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = -a \paren {n - 2... | Primitive of Power of Hyperbolic Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Cosecant_of_a_x | [
"Primitives involving Hyperbolic Cosecant Function"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Derivative of Hyperbolic Cosecant of a x",
"Derivative of Composite Function",
"Primitive of Square of Hyperbolic Cosecant of a x",
"Integration by Parts",
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Linear Combination of Integr... |
proofwiki-9834 | Primitive of Hyperbolic Sine of a x over Power of x | :$\ds \int \frac {\sinh a x \rd x} {x^n} = \frac {-\sinh a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\cosh a x \rd x} {x^{n - 1} }$ | With a view to expressing the problem 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 = \sinh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cosh a x
| c = Derivative of $\sinh a x$
}}... | :$\ds \int \frac {\sinh a x \rd x} {x^n} = \frac {-\sinh a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\cosh a x \rd x} {x^{n - 1} }$ | With a view to expressing the problem 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 = \sinh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cosh a x
| c = [[Derivative of Hyperbolic ... | Primitive of Hyperbolic Sine of a x over Power of x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_over_Power_of_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_over_Power_of_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Derivative of Hyperbolic Sine of a x",
"Primitive of Power",
"Integration by Parts"
] |
proofwiki-9835 | Primitive of Hyperbolic Cosine of a x over Power of x | :$\ds \int \frac {\cosh a x \rd x} {x^n} = \frac {-\cosh a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\sinh a x \rd x} {x^{n - 1} } + C$ | With a view to expressing the problem 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 = \cosh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \sinh a x
| c = Derivative of $\cosh a x$
}}... | :$\ds \int \frac {\cosh a x \rd x} {x^n} = \frac {-\cosh a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\sinh a x \rd x} {x^{n - 1} } + C$ | With a view to expressing the problem 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 = \cosh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \sinh a x
| c = [[Derivative of Hyperbolic ... | Primitive of Hyperbolic Cosine of a x over Power of x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosine_of_a_x_over_Power_of_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosine_of_a_x_over_Power_of_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Derivative of Hyperbolic Cosine of a x",
"Primitive of Power",
"Integration by Parts"
] |
proofwiki-9836 | Primitive of Reciprocal of Power of Hyperbolic Sine of a x | :$\ds \int \frac {\d x} {\sinh^n a x} = \frac {-\cosh a x} {a \paren {n - 1} \sinh^{n - 1} a x} - \frac {n - 2} {n - 1} \int \frac {\d x} {\sinh^{n - 2} a x}$
for $n \ne 1$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh^n a x}
| r = \int \csch^n a x \rd x
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac {-\csch^{n - 2} a x \coth a x} {a \paren {n - 1} } - \frac {n - 2} {n - 1} \int \csch^{n - 2} a x \rd x
| c = Primitive of $\csch^n a x$
}}
{{eqn | ... | :$\ds \int \frac {\d x} {\sinh^n a x} = \frac {-\cosh a x} {a \paren {n - 1} \sinh^{n - 1} a x} - \frac {n - 2} {n - 1} \int \frac {\d x} {\sinh^{n - 2} a x}$
for $n \ne 1$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh^n a x}
| r = \int \csch^n a x \rd x
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac {-\csch^{n - 2} a x \coth a x} {a \paren {n - 1} } - \frac {n - 2} {n - 1} \int \csch^{n - 2} a x \rd x
| c = [[Primitive of Power of Hyperbolic Cos... | Primitive of Reciprocal of Power of Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Primitive of Power of Hyperbolic Cosecant of a x"
] |
proofwiki-9837 | Primitive of Reciprocal of Power of Hyperbolic Cosine of a x | :$\ds \int \frac {\d x} {\cosh^n a x} = \frac {\sinh a x} {a \paren {n - 1} \cosh^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cosh^{n - 2} a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cosh^n a x}
| r = \int \sech^n a x \rd x
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac {\sech^{n - 2} a x \tanh a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sech^{n - 2} a x \rd x + C
| c = Primitive of $\sech^n a x$
}}
{{eqn |... | :$\ds \int \frac {\d x} {\cosh^n a x} = \frac {\sinh a x} {a \paren {n - 1} \cosh^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cosh^{n - 2} a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cosh^n a x}
| r = \int \sech^n a x \rd x
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac {\sech^{n - 2} a x \tanh a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sech^{n - 2} a x \rd x + C
| c = [[Primitive of Power of Hyperbolic Se... | Primitive of Reciprocal of Power of Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Primitive of Power of Hyperbolic Secant of a x"
] |
proofwiki-9838 | Primitive of x over Power of Hyperbolic Sine of a x | :$\ds \int \frac {x \rd x} {\sinh^n a x} = \frac {-x \cosh a x} {a \paren {n - 1} \sinh^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sinh^{n - 2} a x} - \frac {n - 2} {n - 1} \int \frac {x \rd x} {\sinh^{n - 2} a x}$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\sinh^n a x}
| r = \int x \csch^n a x \rd x
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn | r = \frac 1 {a^2} \int \theta \csch^n \theta \rd \theta
| c = Substitution of $a x \to \theta$
}}
{{eqn | ll= \leadsto
| l = \frac 1 {a^2} \int \theta \csch... | :$\ds \int \frac {x \rd x} {\sinh^n a x} = \frac {-x \cosh a x} {a \paren {n - 1} \sinh^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sinh^{n - 2} a x} - \frac {n - 2} {n - 1} \int \frac {x \rd x} {\sinh^{n - 2} a x}$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\sinh^n a x}
| r = \int x \csch^n a x \rd x
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn | r = \frac 1 {a^2} \int \theta \csch^n \theta \rd \theta
| c = [[Integration by Substitution|Substitution of $a x \to \theta$]]
}}
{{eqn | ll= \leadsto
| l =... | Primitive of x over Power of Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_x_over_Power_of_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_over_Power_of_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function"
] | [] | [
"Integration by Substitution",
"Integration by Parts",
"Integration by Parts",
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Primitive of Power of Hyperbolic Cosecant of a x by Hyperbolic Cotangent of a x",
"Integration by Substitution"
] |
proofwiki-9839 | Primitive of x over Power of Hyperbolic Cosine of a x | :$\ds \int \frac {x \rd x} {\cosh^n a x} = \frac {x \sinh a x} {a \paren {n - 1} \cosh^{n - 1} a x} + \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \cosh^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {x \rd x} {\cosh^{n - 2} a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\cosh^n a x}
| r = \int x \sech^n a x \rd x
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \frac 1 {a^2} \int \theta \sech^n \theta \rd \theta
| c = Substitution of $a x \to \theta$
}}
{{eqn | ll= \leadsto
| l = \frac 1 {a^2} \int \theta \sech^n... | :$\ds \int \frac {x \rd x} {\cosh^n a x} = \frac {x \sinh a x} {a \paren {n - 1} \cosh^{n - 1} a x} + \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \cosh^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {x \rd x} {\cosh^{n - 2} a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\cosh^n a x}
| r = \int x \sech^n a x \rd x
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \frac 1 {a^2} \int \theta \sech^n \theta \rd \theta
| c = [[Integration by Substitution|Substitution of $a x \to \theta$]]
}}
{{eqn | ll= \leadsto
| l = \... | Primitive of x over Power of Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_x_over_Power_of_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_over_Power_of_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Integration by Substitution",
"Integration by Parts",
"Integration by Parts",
"Sum of Squares of Hyperbolic Secant and Tangent",
"Primitive of Power of Hyperbolic Secant of a x by Hyperbolic Tangent of a x",
"Integration by Substitution"
] |
proofwiki-9840 | Primitive of Reciprocal of Hyperbolic Cosine of a x plus 1 | :$\ds \int \frac {\d x} {\cosh a x + 1} = \frac 1 a \tanh \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = u
| r = \tanh \frac x 2
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {1 + \cosh x}
| r = \int \frac {\dfrac {2 \rd u} {1 - u^2} } {\dfrac {1 + u^2} {1 - u^2} + 1}
| c = Hyperbolic Tangent Half-Angle Substitution
}}
{{eqn | r = \int \frac {2 \rd u} {1 + u... | :$\ds \int \frac {\d x} {\cosh a x + 1} = \frac 1 a \tanh \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = u
| r = \tanh \frac x 2
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {1 + \cosh x}
| r = \int \frac {\dfrac {2 \rd u} {1 - u^2} } {\dfrac {1 + u^2} {1 - u^2} + 1}
| c = [[Hyperbolic Tangent Half-Angle Substitution]]
}}
{{eqn | r = \int \frac {2 \rd u} {1... | Primitive of Reciprocal of Hyperbolic Cosine of a x plus 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Cosine_of_a_x_plus_1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Cosine_of_a_x_plus_1 | [
"Primitives involving Hyperbolic Cosine Function",
"Hyperbolic Tangent Half-Angle Substitutions"
] | [] | [
"Hyperbolic Tangent Half-Angle Substitution",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Primitive of Constant",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9841 | Primitive of Reciprocal of Hyperbolic Cosine of a x minus 1 | :$\ds \int \frac {\d x} {\cosh a x - 1} = \frac {-1} a \coth \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = u
| r = \tanh \frac x 2
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\cosh x - 1}
| r = \int \frac {\dfrac {2 \rd u} {1 - u^2} } {\dfrac {1 + u^2} {1 - u^2} - 1}
| c = Hyperbolic Tangent Half-Angle Substitution
}}
{{eqn | r = \int \frac {2 \rd u} {1 + u... | :$\ds \int \frac {\d x} {\cosh a x - 1} = \frac {-1} a \coth \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = u
| r = \tanh \frac x 2
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\cosh x - 1}
| r = \int \frac {\dfrac {2 \rd u} {1 - u^2} } {\dfrac {1 + u^2} {1 - u^2} - 1}
| c = [[Hyperbolic Tangent Half-Angle Substitution]]
}}
{{eqn | r = \int \frac {2 \rd u} {1... | Primitive of Reciprocal of Hyperbolic Cosine of a x minus 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Cosine_of_a_x_minus_1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Cosine_of_a_x_minus_1 | [
"Primitives involving Hyperbolic Cosine Function",
"Hyperbolic Tangent Half-Angle Substitutions"
] | [] | [
"Hyperbolic Tangent Half-Angle Substitution",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Primitive of Power",
"Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9842 | Primitive of x over Hyperbolic Cosine of a x plus 1 | :$\ds \int \frac {x \rd x} {\cosh a x + 1} = \frac x a \tanh \frac {a x} 2 - \frac 2 {a^2} \ln \size {\cosh \frac {a x} 2} + C$ | 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 Power
}}
{{end-eqn}}
and let:
... | :$\ds \int \frac {x \rd x} {\cosh a x + 1} = \frac x a \tanh \frac {a x} 2 - \frac 2 {a^2} \ln \size {\cosh \frac {a x} 2} + C$ | 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 Po... | Primitive of x over Hyperbolic Cosine of a x plus 1 | https://proofwiki.org/wiki/Primitive_of_x_over_Hyperbolic_Cosine_of_a_x_plus_1 | https://proofwiki.org/wiki/Primitive_of_x_over_Hyperbolic_Cosine_of_a_x_plus_1 | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Primitive of Reciprocal of Hyperbolic Cosine of a x plus 1",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Hyperbolic Tangent of a x"
] |
proofwiki-9843 | Primitive of x over Hyperbolic Cosine of a x minus 1 | :$\ds \int \frac {x \rd x} {\cosh a x - 1} = -\frac x a \coth \frac {a x} 2 + \frac 2 {a^2} \ln \size {\sinh \frac {a x} 2} + C$ | 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... | :$\ds \int \frac {x \rd x} {\cosh a x - 1} = -\frac x a \coth \frac {a x} 2 + \frac 2 {a^2} \ln \size {\sinh \frac {a x} 2} + C$ | 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 Hyperbolic Cosine of a x minus 1 | https://proofwiki.org/wiki/Primitive_of_x_over_Hyperbolic_Cosine_of_a_x_minus_1 | https://proofwiki.org/wiki/Primitive_of_x_over_Hyperbolic_Cosine_of_a_x_minus_1 | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Reciprocal of Hyperbolic Cosine of a x minus 1",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Hyperbolic Cotangent of a x"
] |
proofwiki-9844 | Primitive of Reciprocal of Square of Hyperbolic Cosine of a x plus 1 | :$\ds \int \frac {\d x} {\paren {\cosh a x + 1}^2} = \frac 1 {2 a} \tanh \frac {a x} 2 - \frac 1 {6 a} \tanh^3 \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {\cosh a x + 1}^2}
| r = \int \paren {\frac 1 2 \sech^2 \frac {a x} 2}^2 \rd x
| c = Reciprocal of Hyperbolic Cosine Plus One
}}
{{eqn | r = \frac 1 4 \int \sech^4 \frac {a x} 2 \rd x
| c = simplifying
}}
{{eqn | r = \frac 1 4 \paren {\frac{\sech^2 \... | :$\ds \int \frac {\d x} {\paren {\cosh a x + 1}^2} = \frac 1 {2 a} \tanh \frac {a x} 2 - \frac 1 {6 a} \tanh^3 \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {\cosh a x + 1}^2}
| r = \int \paren {\frac 1 2 \sech^2 \frac {a x} 2}^2 \rd x
| c = [[Reciprocal of Hyperbolic Cosine Plus One]]
}}
{{eqn | r = \frac 1 4 \int \sech^4 \frac {a x} 2 \rd x
| c = simplifying
}}
{{eqn | r = \frac 1 4 \paren {\frac{\sech... | Primitive of Reciprocal of Square of Hyperbolic Cosine of a x plus 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Hyperbolic_Cosine_of_a_x_plus_1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Hyperbolic_Cosine_of_a_x_plus_1 | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Reciprocal of Hyperbolic Cosine Plus One",
"Primitive of Power of Hyperbolic Secant of a x",
"Primitive of Square of Hyperbolic Secant of a x",
"Sum of Squares of Hyperbolic Secant and Tangent"
] |
proofwiki-9845 | Primitive of Reciprocal of Square of Hyperbolic Cosine of a x minus 1 | :$\ds \int \frac {\d x} {\paren {\cosh a x - 1}^2} = \frac 1 {2 a} \coth \frac {a x} 2 - \frac 1 {6 a} \coth^3 \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {\cosh a x - 1}^2}
| r = \int \paren {\frac 1 2 \csch^2 \frac {a x} 2}^2 \rd x
| c = Reciprocal of Hyperbolic Cosine Minus One
}}
{{eqn | r = \frac 1 4 \int \csch^4 \frac {a x} 2 \rd x
| c = simplifying
}}
{{eqn | r = \frac 1 4 \paren {\frac {-\csch^... | :$\ds \int \frac {\d x} {\paren {\cosh a x - 1}^2} = \frac 1 {2 a} \coth \frac {a x} 2 - \frac 1 {6 a} \coth^3 \frac {a x} 2 + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {\cosh a x - 1}^2}
| r = \int \paren {\frac 1 2 \csch^2 \frac {a x} 2}^2 \rd x
| c = [[Reciprocal of Hyperbolic Cosine Minus One]]
}}
{{eqn | r = \frac 1 4 \int \csch^4 \frac {a x} 2 \rd x
| c = simplifying
}}
{{eqn | r = \frac 1 4 \paren {\frac {-\c... | Primitive of Reciprocal of Square of Hyperbolic Cosine of a x minus 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Hyperbolic_Cosine_of_a_x_minus_1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Hyperbolic_Cosine_of_a_x_minus_1 | [
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Reciprocal of Hyperbolic Cosine Minus One",
"Primitive of Power of Hyperbolic Cosecant of a x",
"Primitive of Square of Hyperbolic Cosecant of a x",
"Difference of Squares of Hyperbolic Cotangent and Cosecant"
] |
proofwiki-9846 | Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x | :$\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh a x \cosh a x \rd x
| r = \int \frac {\sinh 2 a x} 2 \rd x
| c = Double Angle Formula for Hyperbolic Sine
}}
{{eqn | r = \frac 1 2 \int \sinh 2 a x \rd x
| c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \frac 1 2 \paren {\frac {\cosh 2 a x} {2 a} } +... | :$\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh a x \cosh a x \rd x
| r = \int \frac {\sinh 2 a x} 2 \rd x
| c = [[Double Angle Formula for Hyperbolic Sine]]
}}
{{eqn | r = \frac 1 2 \int \sinh 2 a x \rd x
| c = [[Primitive of Constant Multiple of Function]]
}}
{{eqn | r = \frac 1 2 \paren {\frac {\cosh 2 a x} {... | Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x/Proof_1 | [
"Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x",
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Double Angle Formulas/Hyperbolic Sine",
"Primitive of Constant Multiple of Function",
"Primitive of Hyperbolic Sine of a x",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-9847 | Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x | :$\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh a x \cosh a x \rd x
| r = \int \cosh a x \sinh a x \rd x
| c =
}}
{{eqn | r = \frac {\cosh^2 a x} {2 a} + C
| c = Primitive of $\cosh^n a x \sinh a x$ using $n = 1$
}}
{{eqn | r = \frac {1 + \sinh^2 a x} {2 a} + C
| c = Difference of Squares of Hyperbolic Co... | :$\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh a x \cosh a x \rd x
| r = \int \cosh a x \sinh a x \rd x
| c =
}}
{{eqn | r = \frac {\cosh^2 a x} {2 a} + C
| c = [[Primitive of Power of Hyperbolic Cosine of a x by Hyperbolic Sine of a x|Primitive of $\cosh^n a x \sinh a x$]] using $n = 1$
}}
{{eqn | r = \frac {... | Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x/Proof_2 | [
"Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x",
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Primitive of Power of Hyperbolic Cosine of a x by Hyperbolic Sine of a x",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-9848 | Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x | :$\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | n = 1
| l = \int \sinh^n a x \cosh a x \rd x
| r = \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C
| c = Primitive of $\sinh^n a x \cosh a x$
}}
{{eqn | ll= \leadsto
| l = \int \sinh a x \cosh a x \rd x
| r = \frac {\sinh^2 a x} {2 a} + C
| c = setting $n = 1$ ... | :$\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | n = 1
| l = \int \sinh^n a x \cosh a x \rd x
| r = \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C
| c = [[Primitive of Power of Hyperbolic Sine of a x by Hyperbolic Cosine of a x|Primitive of $\sinh^n a x \cosh a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \sinh a x \cosh a x... | Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x/Proof_3 | [
"Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x",
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Primitive of Power of Hyperbolic Sine of a x by Hyperbolic Cosine of a x"
] |
proofwiki-9849 | Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x | :$\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$ | 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 = \sinh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cosh a x
| c = Derivative of $\sinh a x$
... | :$\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$ | With a view to expressing the [[Definition:Primitive (Calculus)|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sinh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cosh a x
... | Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 4 | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x/Proof_4 | [
"Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x",
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Definition:Primitive (Calculus)",
"Derivative of Hyperbolic Sine of a x",
"Primitive of Hyperbolic Cosine of a x",
"Integration by Parts"
] |
proofwiki-9850 | Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x | :$\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = u
| r = \sinh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cosh a x
| c = Derivative of $\sinh a x$
}}
{{eqn | ll= \leadsto
| l = \int \sinh a x \cosh a x \rd x
| r = \int \frac u a \rd u
| c = Integration by Substitution... | :$\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = u
| r = \sinh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cosh a x
| c = [[Derivative of Hyperbolic Sine of a x|Derivative of $\sinh a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \sinh a x \cosh a x \rd x
| r = \int \frac u a \rd ... | Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 5 | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x/Proof_5 | [
"Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x",
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Derivative of Hyperbolic Sine of a x",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9851 | Primitive of Hyperbolic Sine of p x by Hyperbolic Cosine of q x | :$\ds \int \sinh p x \cosh q x \rd x = \frac {\map \cosh {p + q} x} {2 \paren {p + q} } + \frac {\map \cosh {p - q} x} {2 \paren {p - q} } + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh p x \cosh q x \rd x
| r = \int \paren {\frac {\map \sinh {p x + q x} + \map \sinh {p x - q x} } 2} \rd x
| c = Werner Formula for Hyperbolic Sine by Hyperbolic Cosine
}}
{{eqn | r = \frac 1 2 \int \map \sinh {p + q} x \rd x + \frac 1 2 \int \map \sinh {p - q} x \rd x
... | :$\ds \int \sinh p x \cosh q x \rd x = \frac {\map \cosh {p + q} x} {2 \paren {p + q} } + \frac {\map \cosh {p - q} x} {2 \paren {p - q} } + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh p x \cosh q x \rd x
| r = \int \paren {\frac {\map \sinh {p x + q x} + \map \sinh {p x - q x} } 2} \rd x
| c = [[Werner Formula for Hyperbolic Sine by Hyperbolic Cosine]]
}}
{{eqn | r = \frac 1 2 \int \map \sinh {p + q} x \rd x + \frac 1 2 \int \map \sinh {p - q} x \rd x... | Primitive of Hyperbolic Sine of p x by Hyperbolic Cosine of q x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_p_x_by_Hyperbolic_Cosine_of_q_x | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Sine_of_p_x_by_Hyperbolic_Cosine_of_q_x | [
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Werner Formulas/Hyperbolic Sine by Hyperbolic Cosine",
"Linear Combination of Integrals/Indefinite",
"Primitive of Hyperbolic Sine of a x"
] |
proofwiki-9852 | Primitive of Power of Hyperbolic Sine of a x by Hyperbolic Cosine of a x | :$\ds \int \sinh^n a x \cosh a x \rd x = \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C$
for $n \ne -1$. | {{begin-eqn}}
{{eqn | l = u
| r = \sinh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cosh a x
| c = Derivative of $\sinh a x$
}}
{{eqn | ll= \leadsto
| l = \int \sinh^n a x \cosh a x \rd x
| r = \int \frac {u^n} a \rd u
| c = Integration by Substi... | :$\ds \int \sinh^n a x \cosh a x \rd x = \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C$
for $n \ne -1$. | {{begin-eqn}}
{{eqn | l = u
| r = \sinh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \cosh a x
| c = [[Derivative of Hyperbolic Sine of a x|Derivative of $\sinh a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \sinh^n a x \cosh a x \rd x
| r = \int \frac {u^n} ... | Primitive of Power of Hyperbolic Sine of a x by Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Derivative of Hyperbolic Sine of a x",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9853 | Primitive of Power of Hyperbolic Cosine of a x by Hyperbolic Sine of a x | :$\ds \int \cosh^n a x \sinh a x \rd x = \frac {\cosh^{n + 1} a x} {\paren {n + 1} a} + C$
for $n \ne -1$. | {{begin-eqn}}
{{eqn | l = u
| r = \cosh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \sinh a x
| c = Derivative of $\cosh a x$
}}
{{eqn | ll= \leadsto
| l = \int \cosh^n a x \sinh a x \rd x
| r = \int \frac {u^n} a \rd u
| c = Integration by Substi... | :$\ds \int \cosh^n a x \sinh a x \rd x = \frac {\cosh^{n + 1} a x} {\paren {n + 1} a} + C$
for $n \ne -1$. | {{begin-eqn}}
{{eqn | l = u
| r = \cosh a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \sinh a x
| c = [[Derivative of Hyperbolic Cosine of a x|Derivative of $\cosh a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \cosh^n a x \sinh a x \rd x
| r = \int \frac {u^n... | Primitive of Power of Hyperbolic Cosine of a x by Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Cosine_of_a_x_by_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Cosine_of_a_x_by_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Derivative of Hyperbolic Cosine of a x",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9854 | Primitive of Square of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x | :$\ds \int \sinh^2 a x \cosh^2 a x \rd x = \frac {\sinh 4 a x} {32 a} - \frac x 8 + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh^2 a x \cosh^2 a x \rd x
| r = \int \paren {\sinh a x \cosh a x}^2 \rd x
| c =
}}
{{eqn | r = \int \paren {\frac {\sinh 2 a x} 2}^2 \rd x
| c = Double Angle Formula for Hyperbolic Sine
}}
{{eqn | r = \frac 1 4 \int \sinh^2 2 a x \rd x
| c = Primitive of Const... | :$\ds \int \sinh^2 a x \cosh^2 a x \rd x = \frac {\sinh 4 a x} {32 a} - \frac x 8 + C$ | {{begin-eqn}}
{{eqn | l = \int \sinh^2 a x \cosh^2 a x \rd x
| r = \int \paren {\sinh a x \cosh a x}^2 \rd x
| c =
}}
{{eqn | r = \int \paren {\frac {\sinh 2 a x} 2}^2 \rd x
| c = [[Double Angle Formula for Hyperbolic Sine]]
}}
{{eqn | r = \frac 1 4 \int \sinh^2 2 a x \rd x
| c = [[Primitive of... | Primitive of Square of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Sine_of_a_x_by_Square_of_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Sine_of_a_x_by_Square_of_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Double Angle Formulas/Hyperbolic Sine",
"Primitive of Constant Multiple of Function",
"Primitive of Square of Hyperbolic Sine of a x/Corollary"
] |
proofwiki-9855 | Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x | :$\ds \int \frac {\d x} {\sinh a x \cosh a x} = \frac 1 a \ln \size {\tanh a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh a x \cosh a x}
| r = \int \frac {\sech a x \rd x} {\sinh a x}
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \int \frac {\sech^2 a x \rd x} {\sinh a x \sech a x}
| c = multiplying top and bottom by $\sech a x$
}}
{{eqn | r = \int \frac {\s... | :$\ds \int \frac {\d x} {\sinh a x \cosh a x} = \frac 1 a \ln \size {\tanh a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh a x \cosh a x}
| r = \int \frac {\sech a x \rd x} {\sinh a x}
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \int \frac {\sech^2 a x \rd x} {\sinh a x \sech a x}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|b... | Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Primitive of Square of Hyperbolic Secant of a x over Hyperbolic Tangent of a x"
] |
proofwiki-9856 | Primitive of Square of Hyperbolic Secant of a x over Hyperbolic Tangent of a x | :$\ds \int \frac {\sech^2 a x \rd x} {\tanh a x} = \frac 1 a \ln \size {\tanh a x} + C$ | {{begin-eqn}}
{{eqn | l = \frac {\d} {\d x} \tanh a x
| r = a \sech^2 a x
| c = Derivative of $\tanh a x$
}}
{{eqn | ll= \leadsto
| l = \int \frac {a \sech^2 a x \rd x} {\tanh a x}
| r = \ln \size {\tanh a x} + C
| c = Primitive of Function under its Derivative
}}
{{eqn | ll= \leadsto
... | :$\ds \int \frac {\sech^2 a x \rd x} {\tanh a x} = \frac 1 a \ln \size {\tanh a x} + C$ | {{begin-eqn}}
{{eqn | l = \frac {\d} {\d x} \tanh a x
| r = a \sech^2 a x
| c = [[Derivative of Hyperbolic Tangent of a x|Derivative of $\tanh a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {a \sech^2 a x \rd x} {\tanh a x}
| r = \ln \size {\tanh a x} + C
| c = [[Primitive of Function und... | Primitive of Square of Hyperbolic Secant of a x over Hyperbolic Tangent of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Secant_of_a_x_over_Hyperbolic_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Secant_of_a_x_over_Hyperbolic_Tangent_of_a_x | [
"Primitives involving Hyperbolic Tangent Function",
"Primitives involving Hyperbolic Secant Function"
] | [] | [
"Derivative of Hyperbolic Tangent of a x",
"Primitive of Function under its Derivative",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9857 | Primitive of Reciprocal of Square of Hyperbolic Sine of a x by Hyperbolic Cosine of a x | :$\ds \int \frac {\d x} {\sinh^2 a x \cosh a x} = -\frac 1 a \map \arctan {\sinh a x} - \frac {\csch a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh^2 a x \cosh a x}
| r = \int \frac {\paren {\cosh^2 a x - \sinh^2 a x} \rd x} {\sinh^2 a x \cosh a x}
| c = Difference of Squares of Hyperbolic Cosine and Sine
}}
{{eqn | r = \int \frac {\cosh^2 a x \rd x} {\sinh^2 a x \cosh a x} - \int \frac {\sinh^2 a x \r... | :$\ds \int \frac {\d x} {\sinh^2 a x \cosh a x} = -\frac 1 a \map \arctan {\sinh a x} - \frac {\csch a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh^2 a x \cosh a x}
| r = \int \frac {\paren {\cosh^2 a x - \sinh^2 a x} \rd x} {\sinh^2 a x \cosh a x}
| c = [[Difference of Squares of Hyperbolic Cosine and Sine]]
}}
{{eqn | r = \int \frac {\cosh^2 a x \rd x} {\sinh^2 a x \cosh a x} - \int \frac {\sinh^2 a ... | Primitive of Reciprocal of Square of Hyperbolic Sine of a x by Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Difference of Squares of Hyperbolic Cosine and Sine",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power of Hyperbolic Cosecant of a x by Hyperbolic Cotangent of a x",
"Primitive of Hyperbolic Secant of a x/Arctangent of Hyperbolic Sine Form"
] |
proofwiki-9858 | Primitive of Reciprocal of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x | :$\ds \int \frac {\d x} {\sinh a x \cosh^2 a x} = \frac 1 a \ln \size {\tanh \frac {a x} 2} + \frac {\sech a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh^2 a x \cosh a x}
| r = \int \frac {\left({\cosh^2 a x - \sinh^2 a x}\right) \rd x} {\sinh a x \cosh^2 a x}
| c = Difference of Squares of Hyperbolic Cosine and Sine
}}
{{eqn | r = \int \frac {\cosh^2 a x \rd x} {\sinh a x \cosh^2 a x} - \int \frac {\sinh^2 ... | :$\ds \int \frac {\d x} {\sinh a x \cosh^2 a x} = \frac 1 a \ln \size {\tanh \frac {a x} 2} + \frac {\sech a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh^2 a x \cosh a x}
| r = \int \frac {\left({\cosh^2 a x - \sinh^2 a x}\right) \rd x} {\sinh a x \cosh^2 a x}
| c = [[Difference of Squares of Hyperbolic Cosine and Sine]]
}}
{{eqn | r = \int \frac {\cosh^2 a x \rd x} {\sinh a x \cosh^2 a x} - \int \frac {\sin... | Primitive of Reciprocal of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Sine_of_a_x_by_Square_of_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Sine_of_a_x_by_Square_of_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Difference of Squares of Hyperbolic Cosine and Sine",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power of Hyperbolic Secant of a x by Hyperbolic Tangent of a x",
"Primitive of Hyperbolic Cosecant of a x"
] |
proofwiki-9859 | Primitive of Power of Hyperbolic Secant of a x by Hyperbolic Tangent of a x | :$\ds \int \sech^n a x \tanh a x \rd x = \frac {-\sech^n a x} {n a} + C$
for $n \ne 0$. | {{begin-eqn}}
{{eqn | l = z
| r = \sech a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -a \sech a x \tanh a x
| c = Derivative of $\sech a x$
}}
{{eqn | ll= \leadsto
| l = \int \sech^n a x \tanh a x \rd x
| r = \int \frac {-z^{n - 1} \rd z} a
| c = Inte... | :$\ds \int \sech^n a x \tanh a x \rd x = \frac {-\sech^n a x} {n a} + C$
for $n \ne 0$. | {{begin-eqn}}
{{eqn | l = z
| r = \sech a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -a \sech a x \tanh a x
| c = [[Derivative of Hyperbolic Secant of a x|Derivative of $\sech a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \sech^n a x \tanh a x \rd x
| r = \int ... | Primitive of Power of Hyperbolic Secant of a x by Hyperbolic Tangent of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Secant_of_a_x_by_Hyperbolic_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Secant_of_a_x_by_Hyperbolic_Tangent_of_a_x | [
"Primitives involving Hyperbolic Secant Function",
"Primitives involving Hyperbolic Tangent Function"
] | [] | [
"Derivative of Hyperbolic Secant of a x",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9860 | Primitive of Power of Hyperbolic Cosecant of a x by Hyperbolic Cotangent of a x | :$\ds \int \csch^n a x \coth a x \rd x = \frac {-\csch^n a x} {n a} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \csch a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -a \csch a x \coth a x
| c = Derivative of $\csch a x$
}}
{{eqn | ll= \leadsto
| l = \int \csch^n a x \coth a x \rd x
| r = \int \frac {-z^{n - 1} \rd z} a
| c = Inte... | :$\ds \int \csch^n a x \coth a x \rd x = \frac {-\csch^n a x} {n a} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \csch a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -a \csch a x \coth a x
| c = [[Derivative of Hyperbolic Cosecant of a x|Derivative of $\csch a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \csch^n a x \coth a x \rd x
| r = \in... | Primitive of Power of Hyperbolic Cosecant of a x by Hyperbolic Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Cosecant_of_a_x_by_Hyperbolic_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Hyperbolic_Cosecant_of_a_x_by_Hyperbolic_Cotangent_of_a_x | [
"Primitives involving Hyperbolic Cosecant Function",
"Primitives involving Hyperbolic Cotangent Function"
] | [] | [
"Derivative of Hyperbolic Cosecant of a x",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9861 | Primitive of Reciprocal of Square of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x | :$\ds \int \frac {\d x} {\sinh^2 a x \cosh^2 a x} = \frac {-2 \coth 2 a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh^2 a x \cosh^2 a x}
| r = \int \frac {\d x} {\paren {\sinh a x \cosh a x}^2}
| c =
}}
{{eqn | r = \int \frac {\d x} {\paren {\dfrac {\sinh^2 2 a x} 2}^2}
| c = Double Angle Formula for Hyperbolic Sine
}}
{{eqn | r = 4 \int \frac {\d x} {\sinh^2 2 a x}... | :$\ds \int \frac {\d x} {\sinh^2 a x \cosh^2 a x} = \frac {-2 \coth 2 a x} a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh^2 a x \cosh^2 a x}
| r = \int \frac {\d x} {\paren {\sinh a x \cosh a x}^2}
| c =
}}
{{eqn | r = \int \frac {\d x} {\paren {\dfrac {\sinh^2 2 a x} 2}^2}
| c = [[Double Angle Formula for Hyperbolic Sine]]
}}
{{eqn | r = 4 \int \frac {\d x} {\sinh^2 2 ... | Primitive of Reciprocal of Square of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Hyperbolic_Sine_of_a_x_by_Square_of_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Square_of_Hyperbolic_Sine_of_a_x_by_Square_of_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Double Angle Formulas/Hyperbolic Sine",
"Primitive of Constant Multiple of Function",
"Primitive of Square of Hyperbolic Cosecant of a x"
] |
proofwiki-9862 | Primitive of Square of Hyperbolic Sine of a x over Hyperbolic Cosine of a x | :$\ds \int \frac {\sinh^2 a x \rd x} {\cosh a x} = \frac {\sinh a x} a - \frac 1 a \map \arctan {\sinh a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\sinh^2 a x \rd x} {\cosh a x}
| r = \int \frac {\paren {\cosh^2 a x - 1} \rd x} {\cosh a x}
| c = Difference of Squares of Hyperbolic Cosine and Sine
}}
{{eqn | r = \int \frac {\paren {\cosh^2 a x} \rd x} {\cosh a x} - \int \frac {\d x} {\cosh a x}
| c = Linear C... | :$\ds \int \frac {\sinh^2 a x \rd x} {\cosh a x} = \frac {\sinh a x} a - \frac 1 a \map \arctan {\sinh a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\sinh^2 a x \rd x} {\cosh a x}
| r = \int \frac {\paren {\cosh^2 a x - 1} \rd x} {\cosh a x}
| c = [[Difference of Squares of Hyperbolic Cosine and Sine]]
}}
{{eqn | r = \int \frac {\paren {\cosh^2 a x} \rd x} {\cosh a x} - \int \frac {\d x} {\cosh a x}
| c = [[Li... | Primitive of Square of Hyperbolic Sine of a x over Hyperbolic Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Sine_of_a_x_over_Hyperbolic_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Sine_of_a_x_over_Hyperbolic_Cosine_of_a_x | [
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Difference of Squares of Hyperbolic Cosine and Sine",
"Linear Combination of Integrals/Indefinite",
"Primitive of Hyperbolic Cosine of a x",
"Primitive of Hyperbolic Secant of a x/Arctangent of Hyperbolic Sine Form"
] |
proofwiki-9863 | Primitive of Square of Hyperbolic Cosine of a x over Hyperbolic Sine of a x | :$\ds \int \frac {\cosh^2 a x \rd x} {\sinh a x} = \frac {\cosh a x} a + \frac 1 a \ln \size {\tanh \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\cosh^2 a x \rd x} {\sinh a x}
| r = \int \frac {\paren {\sinh^2 a x + 1} \rd x} {\sinh a x}
| c = Difference of Squares of Hyperbolic Cosine and Sine
}}
{{eqn | r = \int \frac {\sinh^2 a x \rd x} {\sinh a x} + \int \frac {\d x} {\sinh a x}
| c = Linear Combinatio... | :$\ds \int \frac {\cosh^2 a x \rd x} {\sinh a x} = \frac {\cosh a x} a + \frac 1 a \ln \size {\tanh \frac {a x} 2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\cosh^2 a x \rd x} {\sinh a x}
| r = \int \frac {\paren {\sinh^2 a x + 1} \rd x} {\sinh a x}
| c = [[Difference of Squares of Hyperbolic Cosine and Sine]]
}}
{{eqn | r = \int \frac {\sinh^2 a x \rd x} {\sinh a x} + \int \frac {\d x} {\sinh a x}
| c = [[Linear Comb... | Primitive of Square of Hyperbolic Cosine of a x over Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cosine_of_a_x_over_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cosine_of_a_x_over_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Difference of Squares of Hyperbolic Cosine and Sine",
"Linear Combination of Integrals/Indefinite",
"Primitive of Hyperbolic Sine of a x",
"Primitive of Hyperbolic Cosecant of a x"
] |
proofwiki-9864 | Primitive of Reciprocal of Hyperbolic Cosine of a x by 1 plus Hyperbolic Sine of a x | :$\ds \int \frac {\rd x} {\cosh a x \paren {1 + \sinh a x} } = \frac 1 {2 a} \ln \size {\frac {1 + \sinh a x} {\cosh a x} } + \frac 1 a \map \arctan {e^{a x} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\rd x} {\cosh a x \paren {1 + \sinh a x} }
| r = \int \frac {\sech^2 a x} {\sech a x + \tanh a x} \rd x
| c = {{Defof|Hyperbolic Cosecant}}, {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \frac 1 2 \int \frac {2 \sech^2 a x} {\sech a x + \tanh a x} \rd x
| c = multiplying and di... | :$\ds \int \frac {\rd x} {\cosh a x \paren {1 + \sinh a x} } = \frac 1 {2 a} \ln \size {\frac {1 + \sinh a x} {\cosh a x} } + \frac 1 a \map \arctan {e^{a x} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\rd x} {\cosh a x \paren {1 + \sinh a x} }
| r = \int \frac {\sech^2 a x} {\sech a x + \tanh a x} \rd x
| c = {{Defof|Hyperbolic Cosecant}}, {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \frac 1 2 \int \frac {2 \sech^2 a x} {\sech a x + \tanh a x} \rd x
| c = multiplying and di... | Primitive of Reciprocal of Hyperbolic Cosine of a x by 1 plus Hyperbolic Sine of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Cosine_of_a_x_by_1_plus_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Cosine_of_a_x_by_1_plus_Hyperbolic_Sine_of_a_x | [
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Hyperbolic Secant of a x/Arctangent of Exponential Form",
"Derivative of Hyperbolic Secant of a x",
"Derivative of Hyperbolic Tangent of a x",
"Primitive of Function under its Derivative"
] |
proofwiki-9865 | Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x plus 1 | :$\ds \int \frac {\d x} {\sinh a x \paren {\cosh a x + 1} } = \frac 1 {2 a} \ln \size {\tanh \frac {a x} 2} + \frac 1 {2 a \paren {\cosh a x + 1} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \cosh a x
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = a \sinh a x
| c = Derivative of $\cosh a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh a x \paren {\cosh a x + 1} }
| r = \int \frac {\sinh a x \rd x} {\sinh^2 a x ... | :$\ds \int \frac {\d x} {\sinh a x \paren {\cosh a x + 1} } = \frac 1 {2 a} \ln \size {\tanh \frac {a x} 2} + \frac 1 {2 a \paren {\cosh a x + 1} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \cosh a x
| c =
}}
{{eqn | l = \frac {\d u} {\d x}
| r = a \sinh a x
| c = [[Derivative of Hyperbolic Cosine of a x|Derivative of $\cosh a x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh a x \paren {\cosh a x + 1} }
| r... | Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x plus 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x_plus_1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x_plus_1 | [
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Derivative of Hyperbolic Cosine of a x",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Integration by Substitution",
"Difference of Two Squares",
"Primitive of Reciprocal of a x + b squared by p x + q",
"Reciprocal of Hyper... |
proofwiki-9866 | Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x minus 1 | :$\ds \int \frac {\d x} {\sinh a x \paren {\cosh a x - 1} } = \frac {-1} {2 a} \ln \size {\tanh \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh a x \paren {\cosh a x - 1} }
| r = \int \frac {\map \sinh {a x} \rd x} {\sinh^2 a x \cosh a x - \sinh^2 a x}
| c = multiplying through $\dfrac {\sinh a x} {\sinh a x}$
}}
{{eqn | r = \int \frac {\map \sinh {a x} \rd x} {\cosh a x \paren {\cosh^2 a x - 1} + 1 - \cos... | :$\ds \int \frac {\d x} {\sinh a x \paren {\cosh a x - 1} } = \frac {-1} {2 a} \ln \size {\tanh \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh a x \paren {\cosh a x - 1} }
| r = \int \frac {\map \sinh {a x} \rd x} {\sinh^2 a x \cosh a x - \sinh^2 a x}
| c = multiplying through $\dfrac {\sinh a x} {\sinh a x}$
}}
{{eqn | r = \int \frac {\map \sinh {a x} \rd x} {\cosh a x \paren {\cosh^2 a x - 1} + 1 - \cos... | Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x minus 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x_minus_1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Hyperbolic_Sine_of_a_x_by_Hyperbolic_Cosine_of_a_x_minus_1 | [
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function"
] | [] | [
"Difference of Squares of Hyperbolic Cosine and Sine",
"Integration by Substitution",
"Definition:Partial Fractions Expansion",
"Primitive of Reciprocal",
"Primitive of Power",
"Double Angle Formulas/Hyperbolic Cosine",
"Logarithm of Power/Natural Logarithm",
"Logarithm of Reciprocal"
] |
proofwiki-9867 | Primitive of Cube of Hyperbolic Tangent of a x | :$\ds \int \tanh^3 a x \rd x = \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \tanh^3 a x \rd x
| r = \int \tanh a x \tanh^2 a x \rd x
| c =
}}
{{eqn | r = \int \tanh a x \paren {1 - \sech^2 a x} \rd x
| c = Sum of Squares of Hyperbolic Secant and Tangent
}}
{{eqn | r = \int \tanh a x \rd x - \int \tanh a x \sech^2 a x \rd x
| c = Linear Co... | :$\ds \int \tanh^3 a x \rd x = \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \tanh^3 a x \rd x
| r = \int \tanh a x \tanh^2 a x \rd x
| c =
}}
{{eqn | r = \int \tanh a x \paren {1 - \sech^2 a x} \rd x
| c = [[Sum of Squares of Hyperbolic Secant and Tangent]]
}}
{{eqn | r = \int \tanh a x \rd x - \int \tanh a x \sech^2 a x \rd x
| c = [[Lin... | Primitive of Cube of Hyperbolic Tangent of a x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Cube_of_Hyperbolic_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Hyperbolic_Tangent_of_a_x/Proof_1 | [
"Primitive of Cube of Hyperbolic Tangent of a x",
"Primitives involving Hyperbolic Tangent Function"
] | [] | [
"Sum of Squares of Hyperbolic Secant and Tangent",
"Linear Combination of Integrals/Indefinite",
"Primitive of Hyperbolic Tangent of a x",
"Primitive of Power of Hyperbolic Tangent of a x by Square of Hyperbolic Secant of a x"
] |
proofwiki-9868 | Primitive of Cube of Hyperbolic Tangent of a x | :$\ds \int \tanh^3 a x \rd x = \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \tanh^3 a x \rd x
| r = -\frac {\tanh^2 a x} {2 a} + \int \tanh a x \rd x
| c = Primitive of Power of $\tanh^n a x$ with $n = 3$
}}
{{eqn | r = \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C
| c = Primitive of $\tanh a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \tanh^3 a x \rd x = \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \tanh^3 a x \rd x
| r = -\frac {\tanh^2 a x} {2 a} + \int \tanh a x \rd x
| c = [[Primitive of Power of Hyperbolic Tangent of a x|Primitive of Power of $\tanh^n a x$]] with $n = 3$
}}
{{eqn | r = \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C
| c = [[Pr... | Primitive of Cube of Hyperbolic Tangent of a x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Cube_of_Hyperbolic_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Hyperbolic_Tangent_of_a_x/Proof_2 | [
"Primitive of Cube of Hyperbolic Tangent of a x",
"Primitives involving Hyperbolic Tangent Function"
] | [] | [
"Primitive of Power of Hyperbolic Tangent of a x",
"Primitive of Hyperbolic Tangent of a x"
] |
proofwiki-9869 | Primitive of Cube of Hyperbolic Cotangent of a x | :$\ds \int \coth^3 a x \rd x = \frac {\ln \size {\sinh a x} } a - \frac {\coth^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \coth^3 a x \rd x
| r = \int \coth a x \coth^2 a x \rd x
| c =
}}
{{eqn | r = \int \coth a x \paren {1 + \csch^2 a x} \rd x
| c = Difference of Squares of Hyperbolic Cotangent and Cosecant
}}
{{eqn | r = \int \coth a x \rd x + \int \coth a x \csch^2 a x \rd x
| c ... | :$\ds \int \coth^3 a x \rd x = \frac {\ln \size {\sinh a x} } a - \frac {\coth^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \coth^3 a x \rd x
| r = \int \coth a x \coth^2 a x \rd x
| c =
}}
{{eqn | r = \int \coth a x \paren {1 + \csch^2 a x} \rd x
| c = [[Difference of Squares of Hyperbolic Cotangent and Cosecant]]
}}
{{eqn | r = \int \coth a x \rd x + \int \coth a x \csch^2 a x \rd x
... | Primitive of Cube of Hyperbolic Cotangent of a x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Cube_of_Hyperbolic_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Hyperbolic_Cotangent_of_a_x/Proof_1 | [
"Primitive of Cube of Hyperbolic Cotangent of a x",
"Primitives involving Hyperbolic Cotangent Function"
] | [] | [
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Linear Combination of Integrals/Indefinite",
"Primitive of Hyperbolic Tangent of a x",
"Primitive of Power of Hyperbolic Cotangent of a x by Square of Hyperbolic Cosecant of a x"
] |
proofwiki-9870 | Primitive of Cube of Hyperbolic Cotangent of a x | :$\ds \int \coth^3 a x \rd x = \frac {\ln \size {\sinh a x} } a - \frac {\coth^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \coth^3 a x \rd x
| r = -\frac {\coth^2 a x} {2 a} + \int \coth a x \rd x
| c = Primitive of Power of $\coth^n a x$ with $n = 3$
}}
{{eqn | r = \frac {\ln \size {\sinh a x} } a - \frac {\coth^2 a x} {2 a} + C
| c = Primitive of $\coth a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \coth^3 a x \rd x = \frac {\ln \size {\sinh a x} } a - \frac {\coth^2 a x} {2 a} + C$ | {{begin-eqn}}
{{eqn | l = \int \coth^3 a x \rd x
| r = -\frac {\coth^2 a x} {2 a} + \int \coth a x \rd x
| c = [[Primitive of Power of Hyperbolic Cotangent of a x|Primitive of Power of $\coth^n a x$]] with $n = 3$
}}
{{eqn | r = \frac {\ln \size {\sinh a x} } a - \frac {\coth^2 a x} {2 a} + C
| c = [[... | Primitive of Cube of Hyperbolic Cotangent of a x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Cube_of_Hyperbolic_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Hyperbolic_Cotangent_of_a_x/Proof_2 | [
"Primitive of Cube of Hyperbolic Cotangent of a x",
"Primitives involving Hyperbolic Cotangent Function"
] | [] | [
"Primitive of Power of Hyperbolic Cotangent of a x",
"Primitive of Hyperbolic Cotangent of a x"
] |
proofwiki-9871 | Primitive of x by Hyperbolic Tangent of a x | :$\ds \int x \tanh a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x}^3} 3 - \frac {\paren {a x}^5} {15} + \frac {2 \paren {a x}^7} {105} + \cdots + \frac { 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$ | {{begin-eqn}}
{{eqn | l = \int x \tanh a x \rd x
| r = \frac 1 {a^2} \int \theta \tanh \theta \rd \theta
| c = Substitution of $a x \to \theta$
}}
{{eqn | r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} \rd \theta
... | :$\ds \int x \tanh a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x}^3} 3 - \frac {\paren {a x}^5} {15} + \frac {2 \paren {a x}^7} {105} + \cdots + \frac { 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$ | {{begin-eqn}}
{{eqn | l = \int x \tanh a x \rd x
| r = \frac 1 {a^2} \int \theta \tanh \theta \rd \theta
| c = [[Integration by Substitution|Substitution of $a x \to \theta$]]
}}
{{eqn | r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, \theta^{2 n - 1... | Primitive of x by Hyperbolic Tangent of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Hyperbolic_Tangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Hyperbolic_Tangent_of_a_x | [
"Primitives involving Hyperbolic Tangent Function"
] | [] | [
"Integration by Substitution",
"Power Series Expansion for Hyperbolic Tangent Function",
"Fubini's Theorem",
"Integration by Substitution"
] |
proofwiki-9872 | Primitive of x by Hyperbolic Cotangent of a x | :$\ds \int x \coth a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} 9 - \frac {\paren {a x}^5} {225} + \cdots + \frac {2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$ | {{begin-eqn}}
{{eqn | l = \int x \coth a x \rd x
| r = \frac 1 {a^2} \int \theta \coth \theta \rd \theta
| c = Substitution of $a x \to \theta$
}}
{{eqn | r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{2^{2 n} B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} \rd \theta
| c = Power Seri... | :$\ds \int x \coth a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} 9 - \frac {\paren {a x}^5} {225} + \cdots + \frac {2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$ | {{begin-eqn}}
{{eqn | l = \int x \coth a x \rd x
| r = \frac 1 {a^2} \int \theta \coth \theta \rd \theta
| c = [[Integration by Substitution|Substitution of $a x \to \theta$]]
}}
{{eqn | r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{2^{2 n} B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} \... | Primitive of x by Hyperbolic Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Hyperbolic_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Hyperbolic_Cotangent_of_a_x | [
"Primitives involving Hyperbolic Cotangent Function"
] | [] | [
"Integration by Substitution",
"Power Series Expansion for Hyperbolic Cotangent Function",
"Fubini's Theorem",
"Integration by Substitution"
] |
proofwiki-9873 | Primitive of Square of Hyperbolic Cosecant of a x over Hyperbolic Cotangent of a x | :$\ds \int \frac {\csch^2 a x \rd x} {\coth a x} = \frac {-1} a \ln \size {\coth a x} + C$ | {{begin-eqn}}
{{eqn | l = \frac \d {\d x} \coth a x
| r = -a \csch^2 a x
| c = Derivative of $\coth a x$
}}
{{eqn | ll= \leadsto
| l = \int \frac {-a \csch^2 a x \rd x} {\coth a x}
| r = \ln \size {\coth a x} + C
| c = Primitive of Function under its Derivative
}}
{{eqn | ll= \leadsto
... | :$\ds \int \frac {\csch^2 a x \rd x} {\coth a x} = \frac {-1} a \ln \size {\coth a x} + C$ | {{begin-eqn}}
{{eqn | l = \frac \d {\d x} \coth a x
| r = -a \csch^2 a x
| c = [[Derivative of Hyperbolic Cotangent of a x|Derivative of $\coth a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {-a \csch^2 a x \rd x} {\coth a x}
| r = \ln \size {\coth a x} + C
| c = [[Primitive of Function u... | Primitive of Square of Hyperbolic Cosecant of a x over Hyperbolic Cotangent of a x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cosecant_of_a_x_over_Hyperbolic_Cotangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Square_of_Hyperbolic_Cosecant_of_a_x_over_Hyperbolic_Cotangent_of_a_x | [
"Primitives involving Hyperbolic Cotangent Function",
"Primitives involving Hyperbolic Cosecant Function"
] | [] | [
"Derivative of Hyperbolic Cotangent of a x",
"Primitive of Function under its Derivative",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9874 | Primitive of Cube of Hyperbolic Secant of a x | :$\ds \int \sech^3 a x \rd x = \frac {\sech a x \tanh a x} {2 a} + \frac 1 {2 a} \map \arctan {\sinh a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \sech^3 x \rd x
| r = \frac {\sech a x \tanh a x} {2 a} + \frac 1 2 \int \sech a x \rd x
| c = Primitive of $\sech^n a x$ where $n = 3$
}}
{{eqn | r = \frac {\sech a x \tanh a x} {2 a} + \frac 1 {2 a} \map \arctan {\sinh a x} + C
| c = Primitive of $\sech a x$
}}
{{end-e... | :$\ds \int \sech^3 a x \rd x = \frac {\sech a x \tanh a x} {2 a} + \frac 1 {2 a} \map \arctan {\sinh a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \sech^3 x \rd x
| r = \frac {\sech a x \tanh a x} {2 a} + \frac 1 2 \int \sech a x \rd x
| c = [[Primitive of Power of Hyperbolic Secant of a x|Primitive of $\sech^n a x$]] where $n = 3$
}}
{{eqn | r = \frac {\sech a x \tanh a x} {2 a} + \frac 1 {2 a} \map \arctan {\sinh a x} ... | Primitive of Cube of Hyperbolic Secant of a x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Hyperbolic_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Hyperbolic_Secant_of_a_x | [
"Primitives involving Hyperbolic Secant Function"
] | [] | [
"Primitive of Power of Hyperbolic Secant of a x",
"Primitive of Hyperbolic Secant of a x/Arctangent of Hyperbolic Sine Form"
] |
proofwiki-9875 | Primitive of Cube of Hyperbolic Cosecant of a x | :$\ds \int \csch^3 a x \rd x = \frac {-\csch a x \coth a x} {2 a} - \frac 1 {2 a} \ln \size {\tanh a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \csch^3 x \rd x
| r = \frac {\csch a x \coth a x} {2 a} - \frac 1 2 \int \csch a x \rd x
| c = Primitive of $\csch^n a x$ where $n = 3$
}}
{{eqn | r = \frac {\csch a x \coth a x} {2 a} - \ln \size {\tanh a x} + C
| c = Primitive of $\csch a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \csch^3 a x \rd x = \frac {-\csch a x \coth a x} {2 a} - \frac 1 {2 a} \ln \size {\tanh a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \csch^3 x \rd x
| r = \frac {\csch a x \coth a x} {2 a} - \frac 1 2 \int \csch a x \rd x
| c = [[Primitive of Power of Hyperbolic Cosecant of a x|Primitive of $\csch^n a x$]] where $n = 3$
}}
{{eqn | r = \frac {\csch a x \coth a x} {2 a} - \ln \size {\tanh a x} + C
| c =... | Primitive of Cube of Hyperbolic Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Hyperbolic_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cube_of_Hyperbolic_Cosecant_of_a_x | [
"Primitives involving Hyperbolic Cosecant Function"
] | [] | [
"Primitive of Power of Hyperbolic Cosecant of a x",
"Primitive of Hyperbolic Cosecant of a x"
] |
proofwiki-9876 | Primitive of Hyperbolic Secant Function/Arctangent of Hyperbolic Sine Form | :$\ds \int \sech x \rd x = \map \arctan {\sinh x} + C$ | We have that:
{{begin-eqn}}
{{eqn | l = \int \sech x \rd x
| r = \int \frac {\d x} {\cosh x}
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \int \frac {\cosh x \rd x} {\cosh^2 x}
| c = multiplying top and bottom by $\cosh x$
}}
{{eqn | r = \int \frac {\cosh x \rd x} {1 + \sinh^2 x}
|... | :$\ds \int \sech x \rd x = \map \arctan {\sinh x} + C$ | We have that:
{{begin-eqn}}
{{eqn | l = \int \sech x \rd x
| r = \int \frac {\d x} {\cosh x}
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \int \frac {\cosh x \rd x} {\cosh^2 x}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $\cosh x$
}}
{{eqn | r ... | Primitive of Hyperbolic Secant Function/Arctangent of Hyperbolic Sine Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_Function/Arctangent_of_Hyperbolic_Sine_Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_Function/Arctangent_of_Hyperbolic_Sine_Form | [
"Primitive of Hyperbolic Secant Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Derivative of Hyperbolic Sine",
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-9877 | Primitive of Hyperbolic Secant of a x/Arcsine Form | :$\ds \int \sech a x \rd x = \frac {\map \arcsin {\tanh a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \sech x \rd x
| r = \map \arcsin {\tanh x}
| c = Primitive of $\sech x$: Arcsine form
}}
{{eqn | ll= \leadsto
| l = \int \sech a x \rd x
| r = \frac 1 a \map \arcsin {\tanh a x} + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\map \... | :$\ds \int \sech a x \rd x = \frac {\map \arcsin {\tanh a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \sech x \rd x
| r = \map \arcsin {\tanh x}
| c = [[Primitive of Hyperbolic Secant Function/Arcsine Form|Primitive of $\sech x$: Arcsine form]]
}}
{{eqn | ll= \leadsto
| l = \int \sech a x \rd x
| r = \frac 1 a \map \arcsin {\tanh a x} + C
| c = [[Primitive of... | Primitive of Hyperbolic Secant of a x/Arcsine Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_of_a_x/Arcsine_Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_of_a_x/Arcsine_Form | [
"Primitives involving Hyperbolic Secant Function"
] | [] | [
"Primitive of Hyperbolic Secant Function/Arcsine Form",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9878 | Primitive of Hyperbolic Secant of a x/Arctangent of Exponential Form | :$\ds \int \sech a x \rd x = \frac {2 \map \arctan {e^{a x} } } a + C$ | {{begin-eqn}}
{{eqn | l = \int \sech x \rd x
| r = 2 \map \arctan {e^x} + C
| c = Primitive of $\sech x$: Arctangent of Exponential Form
}}
{{eqn | ll= \leadsto
| l = \int \sech a x \rd x
| r = \frac 1 a \paren {2 \map \arctan {e^{a x} } } + C
| c = Primitive of Function of Constant Multip... | :$\ds \int \sech a x \rd x = \frac {2 \map \arctan {e^{a x} } } a + C$ | {{begin-eqn}}
{{eqn | l = \int \sech x \rd x
| r = 2 \map \arctan {e^x} + C
| c = [[Primitive of Hyperbolic Secant Function/Arctangent of Exponential Form|Primitive of $\sech x$: Arctangent of Exponential Form]]
}}
{{eqn | ll= \leadsto
| l = \int \sech a x \rd x
| r = \frac 1 a \paren {2 \map \a... | Primitive of Hyperbolic Secant of a x/Arctangent of Exponential Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_of_a_x/Arctangent_of_Exponential_Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_of_a_x/Arctangent_of_Exponential_Form | [
"Primitives involving Hyperbolic Secant Function"
] | [] | [
"Primitive of Hyperbolic Secant Function/Arctangent of Exponential Form",
"Primitive of Function of Constant Multiple"
] |
proofwiki-9879 | Primitive of Hyperbolic Secant of a x/Arctangent of Hyperbolic Sine Form | :$\ds \int \sech a x \rd x = \frac {\map \arctan {\sinh a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \sech x \rd x
| r = \map \arctan {\sinh x}
| c = Primitive of $\sech x$: Arctangent of Hyperbolic Sine form
}}
{{eqn | ll= \leadsto
| l = \int \sech a x \rd x
| r = \frac 1 a \map \arctan {\sinh a x} + C
| c = Primitive of Function of Constant Multiple
}}
{{e... | :$\ds \int \sech a x \rd x = \frac {\map \arctan {\sinh a x} } a + C$ | {{begin-eqn}}
{{eqn | l = \int \sech x \rd x
| r = \map \arctan {\sinh x}
| c = [[Primitive of Hyperbolic Secant Function/Arctangent of Hyperbolic Sine Form|Primitive of $\sech x$: Arctangent of Hyperbolic Sine form]]
}}
{{eqn | ll= \leadsto
| l = \int \sech a x \rd x
| r = \frac 1 a \map \arcta... | Primitive of Hyperbolic Secant of a x/Arctangent of Hyperbolic Sine Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_of_a_x/Arctangent_of_Hyperbolic_Sine_Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_of_a_x/Arctangent_of_Hyperbolic_Sine_Form | [
"Primitives involving Hyperbolic Secant Function"
] | [] | [
"Primitive of Hyperbolic Secant Function/Arctangent of Hyperbolic Sine Form",
"Primitive of Function of Constant Multiple",
"Category:Primitives involving Hyperbolic Secant Function"
] |
proofwiki-9880 | Primitive of x by Hyperbolic Secant of a x | {{begin-eqn}}
{{eqn | l = \int x \sech a x \rd x
| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C
}}
{{eqn | r = \frac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 - \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} - \cdots } + C
}}
... | {{begin-eqn}}
{{eqn | l = \int x \sech a x \rd x
| r = \frac 1 {a^2} \int \theta \sech \theta \rd \theta
| c = Substitution of $a x \to \theta$
}}
{{eqn | r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{E_{2 n} \, \theta^{2 n} } {\paren {2 n}!} \rd \theta
| c = Power Series Expansion for Hyperbol... | {{begin-eqn}}
{{eqn | l = \int x \sech a x \rd x
| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C
}}
{{eqn | r = \frac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 - \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} - \cdots } + C
}}
... | {{begin-eqn}}
{{eqn | l = \int x \sech a x \rd x
| r = \frac 1 {a^2} \int \theta \sech \theta \rd \theta
| c = [[Integration by Substitution|Substitution of $a x \to \theta$]]
}}
{{eqn | r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{E_{2 n} \, \theta^{2 n} } {\paren {2 n}!} \rd \theta
| c = [[P... | Primitive of x by Hyperbolic Secant of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Hyperbolic_Secant_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Hyperbolic_Secant_of_a_x | [
"Primitives involving Hyperbolic Secant Function"
] | [] | [
"Integration by Substitution",
"Power Series Expansion for Hyperbolic Secant Function",
"Fubini's Theorem",
"Integration by Substitution"
] |
proofwiki-9881 | Primitive of x by Hyperbolic Cosecant of a x | {{begin-eqn}}
{{eqn | l = \int x \csch a x \rd x
| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C
}}
{{eqn | r = \dfrac 1 {a^2} \paren {a x - \dfrac {\paren {a x}^3} {18} + \dfrac {7 \paren {a x}^5} {1800} - \cdots} + C
}}
... | {{begin-eqn}}
{{eqn | l = \int x \csch a x \rd x
| r = \frac 1 {a^2} \int \theta \csch \theta \rd \theta
| c = Substitution of $a x \to \theta$
}}
{{eqn | r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} \rd \theta
| c = Po... | {{begin-eqn}}
{{eqn | l = \int x \csch a x \rd x
| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C
}}
{{eqn | r = \dfrac 1 {a^2} \paren {a x - \dfrac {\paren {a x}^3} {18} + \dfrac {7 \paren {a x}^5} {1800} - \cdots} + C
}}
... | {{begin-eqn}}
{{eqn | l = \int x \csch a x \rd x
| r = \frac 1 {a^2} \int \theta \csch \theta \rd \theta
| c = [[Integration by Substitution|Substitution of $a x \to \theta$]]
}}
{{eqn | r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \, \theta^{2 n - 1} } {\pa... | Primitive of x by Hyperbolic Cosecant of a x | https://proofwiki.org/wiki/Primitive_of_x_by_Hyperbolic_Cosecant_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_by_Hyperbolic_Cosecant_of_a_x | [
"Primitives involving Hyperbolic Cosecant Function"
] | [
"Definition:Bernoulli Numbers"
] | [
"Integration by Substitution",
"Power Series Expansion for Hyperbolic Cosecant Function",
"Fubini's Theorem",
"Integration by Substitution"
] |
proofwiki-9882 | Derivative of Real Area Hyperbolic Sine of x over a | :$\dfrac {\map \d {\map \arsinh {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {x^2 + a^2}}$ | {{begin-eqn}}
{{eqn | l = \frac {\map \d {\map \arsinh {\frac x a} } } {\d x}
| r = \frac 1 a \frac 1 {\sqrt {\paren {\frac x a}^2} + 1}
| c = Derivative of $\arsinh$ and Derivative of Function of Constant Multiple
}}
{{eqn | r = \frac 1 a \frac 1 {\sqrt {\frac {x^2 + a^2} {a^2} } }
| c =
}}
{{eqn |... | :$\dfrac {\map \d {\map \arsinh {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {x^2 + a^2}}$ | {{begin-eqn}}
{{eqn | l = \frac {\map \d {\map \arsinh {\frac x a} } } {\d x}
| r = \frac 1 a \frac 1 {\sqrt {\paren {\frac x a}^2} + 1}
| c = [[Derivative of Inverse Hyperbolic Sine|Derivative of $\arsinh$]] and [[Derivative of Function of Constant Multiple]]
}}
{{eqn | r = \frac 1 a \frac 1 {\sqrt {\frac... | Derivative of Real Area Hyperbolic Sine of x over a | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Sine_of_x_over_a | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Sine_of_x_over_a | [
"Derivative of Inverse Hyperbolic Sine"
] | [] | [
"Derivative of Inverse Hyperbolic Sine",
"Derivative of Function of Constant Multiple"
] |
proofwiki-9883 | Derivative of Real Area Hyperbolic Cosine of x over a | :$\dfrac {\map \d {\map \arcosh {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {x^2 - a^2} }$
where $x > a$. | Let $x > a$.
Then $\dfrac x a > 1$ and so:
{{begin-eqn}}
{{eqn | l = \frac {\map \d {\map {\cosh^{-1} } {\frac x a} } } {\d x}
| r = \frac 1 a \frac 1 {\sqrt {\paren {\frac x a}^2} - 1}
| c = Derivative of $\arcosh$ and Derivative of Function of Constant Multiple
}}
{{eqn | r = \frac 1 a \frac 1 {\sqrt {\f... | :$\dfrac {\map \d {\map \arcosh {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {x^2 - a^2} }$
where $x > a$. | Let $x > a$.
Then $\dfrac x a > 1$ and so:
{{begin-eqn}}
{{eqn | l = \frac {\map \d {\map {\cosh^{-1} } {\frac x a} } } {\d x}
| r = \frac 1 a \frac 1 {\sqrt {\paren {\frac x a}^2} - 1}
| c = [[Derivative of Real Area Hyperbolic Cosine|Derivative of $\arcosh$]] and [[Derivative of Function of Constant Mul... | Derivative of Real Area Hyperbolic Cosine of x over a | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cosine_of_x_over_a | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cosine_of_x_over_a | [
"Derivative of Real Area Hyperbolic Cosine"
] | [] | [
"Derivative of Real Area Hyperbolic Cosine",
"Derivative of Function of Constant Multiple",
"Definition:Real Function/Domain"
] |
proofwiki-9884 | Derivative of Real Area Hyperbolic Tangent of x over a | :$\map {\dfrac \d {\d x} } {\map \artanh {\dfrac x a} } = \dfrac a {a^2 - x^2}$
where $-a < x < a$. | Let $-a < x < a$.
Then $-1 < \dfrac x a < 1$ and so:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\map \artanh {\dfrac x a} }
| r = \frac 1 a \frac 1 {1 - \paren {\frac x a}^2}
| c = Derivative of $\artanh$ and Derivative of Function of Constant Multiple
}}
{{eqn | r = \frac 1 a \frac 1 {\frac {a^2 ... | :$\map {\dfrac \d {\d x} } {\map \artanh {\dfrac x a} } = \dfrac a {a^2 - x^2}$
where $-a < x < a$. | Let $-a < x < a$.
Then $-1 < \dfrac x a < 1$ and so:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\map \artanh {\dfrac x a} }
| r = \frac 1 a \frac 1 {1 - \paren {\frac x a}^2}
| c = [[Derivative of Inverse Hyperbolic Tangent|Derivative of $\artanh$]] and [[Derivative of Function of Constant Multip... | Derivative of Real Area Hyperbolic Tangent of x over a | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Tangent_of_x_over_a | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Tangent_of_x_over_a | [
"Derivative of Inverse Hyperbolic Tangent"
] | [] | [
"Derivative of Inverse Hyperbolic Tangent",
"Derivative of Function of Constant Multiple"
] |
proofwiki-9885 | Derivative of Real Area Hyperbolic Cotangent of x over a | :$\map {\dfrac \d {\d x} } {\map \arcoth {\dfrac x a} } = \dfrac {-a} {x^2 - a^2}$
where $x^2 > a^2$. | Let $x^2 > a^2$.
Then either $\dfrac x a < -1$ or $\dfrac x a > 1$ and so:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\map \arcoth {\dfrac x a} }
| r = \frac 1 a \frac 1 {1 - \paren {\frac x a}^2}
| c = Derivative of $\arcoth$ and Derivative of Function of Constant Multiple
}}
{{eqn | r = \frac 1 ... | :$\map {\dfrac \d {\d x} } {\map \arcoth {\dfrac x a} } = \dfrac {-a} {x^2 - a^2}$
where $x^2 > a^2$. | Let $x^2 > a^2$.
Then either $\dfrac x a < -1$ or $\dfrac x a > 1$ and so:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\map \arcoth {\dfrac x a} }
| r = \frac 1 a \frac 1 {1 - \paren {\frac x a}^2}
| c = [[Derivative of Inverse Hyperbolic Cotangent|Derivative of $\arcoth$]] and [[Derivative of Fun... | Derivative of Real Area Hyperbolic Cotangent of x over a | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cotangent_of_x_over_a | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cotangent_of_x_over_a | [
"Derivative of Inverse Hyperbolic Cotangent"
] | [] | [
"Derivative of Inverse Hyperbolic Cotangent",
"Derivative of Function of Constant Multiple"
] |
proofwiki-9886 | Derivative of Real Area Hyperbolic Secant of x over a | :$\dfrac {\map \d {\map \arsech {\frac x a} } } {\d x} = \dfrac {-a} {x \sqrt{a^2 - x^2} }$
where $0 < x < a$. | Let $0 < x < a$.
Then $0 < \dfrac x a < 1$ and so:
{{begin-eqn}}
{{eqn | l = \frac {\map \rd {\map \arsech {\frac x a} } } {\rd x}
| r = \frac 1 a \frac {-1} {\frac x a \sqrt {1 - \paren {\frac x a}^2} }
| c = Derivative of $\arsech$ and Derivative of Function of Constant Multiple
}}
{{eqn | r = \frac 1 a ... | :$\dfrac {\map \d {\map \arsech {\frac x a} } } {\d x} = \dfrac {-a} {x \sqrt{a^2 - x^2} }$
where $0 < x < a$. | Let $0 < x < a$.
Then $0 < \dfrac x a < 1$ and so:
{{begin-eqn}}
{{eqn | l = \frac {\map \rd {\map \arsech {\frac x a} } } {\rd x}
| r = \frac 1 a \frac {-1} {\frac x a \sqrt {1 - \paren {\frac x a}^2} }
| c = [[Derivative of Inverse Hyperbolic Secant|Derivative of $\arsech$]] and [[Derivative of Function... | Derivative of Real Area Hyperbolic Secant of x over a | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Secant_of_x_over_a | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Secant_of_x_over_a | [
"Derivatives of Inverse Hyperbolic Functions",
"Inverse Hyperbolic Secant"
] | [] | [
"Derivative of Inverse Hyperbolic Secant",
"Derivative of Function of Constant Multiple"
] |
proofwiki-9887 | Derivative of Real Area Hyperbolic Cosecant of x over a | :$\dfrac {\map \d {\arcsch \dfrac x a} } {\d x} = \dfrac {-a} {\size x \sqrt {a^2 + x^2} }$
where $x \ne 0$. | Let $0 < x < a$.
Then $0 < \dfrac x a < 1$ and so:
{{begin-eqn}}
{{eqn | l = \frac {\map \d {\arcsch \dfrac x a} } {\d x}
| r = \frac 1 a \dfrac {-1} {\size {\frac x a} \sqrt {1 + \paren {\frac x a}^2} }
| c = Derivative of $\arcsch$ and Derivative of Function of Constant Multiple
}}
{{eqn | r = \frac 1 a ... | :$\dfrac {\map \d {\arcsch \dfrac x a} } {\d x} = \dfrac {-a} {\size x \sqrt {a^2 + x^2} }$
where $x \ne 0$. | Let $0 < x < a$.
Then $0 < \dfrac x a < 1$ and so:
{{begin-eqn}}
{{eqn | l = \frac {\map \d {\arcsch \dfrac x a} } {\d x}
| r = \frac 1 a \dfrac {-1} {\size {\frac x a} \sqrt {1 + \paren {\frac x a}^2} }
| c = [[Derivative of Inverse Hyperbolic Cosecant|Derivative of $\arcsch$]] and [[Derivative of Functi... | Derivative of Real Area Hyperbolic Cosecant of x over a | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cosecant_of_x_over_a | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cosecant_of_x_over_a | [
"Derivatives of Inverse Hyperbolic Functions",
"Inverse Hyperbolic Cosecant"
] | [] | [
"Derivative of Inverse Hyperbolic Cosecant",
"Derivative of Function of Constant Multiple"
] |
proofwiki-9888 | Primitive of Inverse Hyperbolic Secant of x over a | :$\ds \int \arsech \frac x a \rd x = x \arsech \dfrac x a + a \arcsin \dfrac x a + C$ | 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 = \arsech \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x \sqrt {a^2 - x^2} }
| ... | :$\ds \int \arsech \frac x a \rd x = x \arsech \dfrac x a + a \arcsin \dfrac x a + C$ | 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 = \arsech \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x \s... | Primitive of Inverse Hyperbolic Secant of x over a | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Secant_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Secant_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Secant Function",
"Primitive of Inverse Hyperbolic Secant of x over a"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Secant of x over a",
"Primitive of Constant",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form"
] |
proofwiki-9889 | Primitive of Inverse Hyperbolic Cosecant of x over a | :<nowiki>$\ds \int \arcsch \frac x a \rd x = \begin {cases}
x \arcsch \dfrac x a + a \arsinh \dfrac x a + C & : x > 0 \\ \\
x \arcsch \dfrac x a - a \arsinh \dfrac x a + C & : x < 0
\end {cases}$</nowiki> | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsch \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {\size x \sqrt {a^2 + x^2} }
... | :<nowiki>$\ds \int \arcsch \frac x a \rd x = \begin {cases}
x \arcsch \dfrac x a + a \arsinh \dfrac x a + C & : x > 0 \\ \\
x \arcsch \dfrac x a - a \arsinh \dfrac x a + C & : x < 0
\end {cases}$</nowiki> | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsch \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {\siz... | Primitive of Inverse Hyperbolic Cosecant of x over a | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Cosecant_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Cosecant_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Cosecant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Cosecant of x over a",
"Primitive of Constant",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of Root of x squared plus a squared/Inverse Hyperbolic Sine Form"
] |
proofwiki-9890 | Primitive of x by Inverse Hyperbolic Sine of x over a | :$\ds \int x \arsinh \frac x a \rd x = \paren {\frac {x^2} 2 + \frac {a^2} 4} \arsinh \frac x a - \frac {x \sqrt {x^2 + a^2} } 4 + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arsinh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {x^2 + a^2} }
| c = D... | :$\ds \int x \arsinh \frac x a \rd x = \paren {\frac {x^2} 2 + \frac {a^2} 4} \arsinh \frac x a - \frac {x \sqrt {x^2 + a^2} } 4 + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arsinh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {... | Primitive of x by Inverse Hyperbolic Sine of x over a | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Sine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Sine_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Sine of x over a",
"Primitive of Power",
"Integration by Parts",
"Primitive of x squared over Root of x squared plus a squared/Inverse Hyperbolic Sine Form"
] |
proofwiki-9891 | Real Area Hyperbolic Cosine of x over a in Logarithm Form | :$\arcosh \dfrac x a = \map \ln {x + \sqrt {x^2 - a^2} } - \ln a$ | {{begin-eqn}}
{{eqn | l = \arcosh \frac x a
| r = \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 - 1} }
| c = {{Defof|Real Area Hyperbolic Cosine}}
}}
{{eqn | r = \map \ln {\frac x a + \sqrt {\frac {x^2 - a^2} {a^2} } }
| c =
}}
{{eqn | r = \map \ln {\frac x a + \frac {\sqrt {x^2 - a^2} } a}
... | :$\arcosh \dfrac x a = \map \ln {x + \sqrt {x^2 - a^2} } - \ln a$ | {{begin-eqn}}
{{eqn | l = \arcosh \frac x a
| r = \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 - 1} }
| c = {{Defof|Real Area Hyperbolic Cosine}}
}}
{{eqn | r = \map \ln {\frac x a + \sqrt {\frac {x^2 - a^2} {a^2} } }
| c =
}}
{{eqn | r = \map \ln {\frac x a + \frac {\sqrt {x^2 - a^2} } a}
... | Real Area Hyperbolic Cosine of x over a in Logarithm Form | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Cosine_of_x_over_a_in_Logarithm_Form | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Cosine_of_x_over_a_in_Logarithm_Form | [
"Inverse Hyperbolic Cosine"
] | [] | [
"Difference of Logarithms"
] |
proofwiki-9892 | Real Area Hyperbolic Tangent of x over a in Logarithm Form | :$\artanh \dfrac x a = \dfrac 1 2 \map \ln {\dfrac {a + x} {a - x} }$ | {{begin-eqn}}
{{eqn | l = \artanh \frac x a
| r = \frac 1 2 \map \ln {\frac {1 + \frac x a} {1 - \frac x a} }
| c = {{Defof|Real Area Hyperbolic Tangent}}
}}
{{eqn | r = \frac 1 2 \map \ln {\frac {a + x} {a - x} }
| c = multiplying top and bottom by $a$
}}
{{end-eqn}}
{{qed}} | :$\artanh \dfrac x a = \dfrac 1 2 \map \ln {\dfrac {a + x} {a - x} }$ | {{begin-eqn}}
{{eqn | l = \artanh \frac x a
| r = \frac 1 2 \map \ln {\frac {1 + \frac x a} {1 - \frac x a} }
| c = {{Defof|Real Area Hyperbolic Tangent}}
}}
{{eqn | r = \frac 1 2 \map \ln {\frac {a + x} {a - x} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by... | Real Area Hyperbolic Tangent of x over a in Logarithm Form | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Tangent_of_x_over_a_in_Logarithm_Form | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Tangent_of_x_over_a_in_Logarithm_Form | [
"Inverse Hyperbolic Tangent"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-9893 | Real Area Hyperbolic Cotangent of x over a in Logarithm Form | :$\arcoth \dfrac x a = \dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} }$ | {{begin-eqn}}
{{eqn | l = \arcoth \frac x a
| r = \frac 1 2 \map \ln {\frac {\frac x a + 1} {\frac x a - 1} }
| c = {{Defof|Real Area Hyperbolic Cotangent}}
}}
{{eqn | r = \frac 1 2 \map \ln {\frac {x + a} {x - a} }
| c = multiplying top and bottom by $a$
}}
{{end-eqn}}
{{qed}} | :$\arcoth \dfrac x a = \dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} }$ | {{begin-eqn}}
{{eqn | l = \arcoth \frac x a
| r = \frac 1 2 \map \ln {\frac {\frac x a + 1} {\frac x a - 1} }
| c = {{Defof|Real Area Hyperbolic Cotangent}}
}}
{{eqn | r = \frac 1 2 \map \ln {\frac {x + a} {x - a} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] b... | Real Area Hyperbolic Cotangent of x over a in Logarithm Form | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Cotangent_of_x_over_a_in_Logarithm_Form | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Cotangent_of_x_over_a_in_Logarithm_Form | [
"Inverse Hyperbolic Cotangent"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-9894 | Real Area Hyperbolic Secant of x over a in Logarithm Form | :$\arsech \dfrac x a = \map \ln {\dfrac {a + \sqrt {a^2 - x^2} } x}$ | {{begin-eqn}}
{{eqn | l = \arsech \frac x a
| r = \map \ln {\frac {1 + \sqrt {1 - \paren {\frac x a}^2} } {\frac x a} }
| c = {{Defof|Real Area Hyperbolic Secant}}
}}
{{eqn | r = \map \ln {\frac {a \paren {1 + \sqrt {\dfrac {a^2 - x^2} {a^2} } } } x}
| c =
}}
{{eqn | r = \map \ln {\frac {a \paren {\d... | :$\arsech \dfrac x a = \map \ln {\dfrac {a + \sqrt {a^2 - x^2} } x}$ | {{begin-eqn}}
{{eqn | l = \arsech \frac x a
| r = \map \ln {\frac {1 + \sqrt {1 - \paren {\frac x a}^2} } {\frac x a} }
| c = {{Defof|Real Area Hyperbolic Secant}}
}}
{{eqn | r = \map \ln {\frac {a \paren {1 + \sqrt {\dfrac {a^2 - x^2} {a^2} } } } x}
| c =
}}
{{eqn | r = \map \ln {\frac {a \paren {\d... | Real Area Hyperbolic Secant of x over a in Logarithm Form | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Secant_of_x_over_a_in_Logarithm_Form | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Secant_of_x_over_a_in_Logarithm_Form | [
"Inverse Hyperbolic Secant"
] | [] | [] |
proofwiki-9895 | Real Area Hyperbolic Cosecant of x over a in Logarithm Form | For $a > 0$:
:$\arcsch \dfrac x a = \map \ln {\dfrac {a + \sqrt {a^2 + x^2} } {\size x} }$ | We have that $\arcsch \dfrac x a$ is defined whenever $x \ne 0$.
{{begin-eqn}}
{{eqn | l = \arcsch \frac x a
| r = \map \ln {\frac 1 {x/a} + \frac {\sqrt {1 + \paren {\frac x a}^2} } {\size {\frac x a} } }
| c = {{Defof|Real Area Hyperbolic Cosecant}}
}}
{{eqn | r = \map \ln {\frac a x + \frac {a \paren {\s... | For $a > 0$:
:$\arcsch \dfrac x a = \map \ln {\dfrac {a + \sqrt {a^2 + x^2} } {\size x} }$ | We have that $\arcsch \dfrac x a$ is defined whenever $x \ne 0$.
{{begin-eqn}}
{{eqn | l = \arcsch \frac x a
| r = \map \ln {\frac 1 {x/a} + \frac {\sqrt {1 + \paren {\frac x a}^2} } {\size {\frac x a} } }
| c = {{Defof|Real Area Hyperbolic Cosecant}}
}}
{{eqn | r = \map \ln {\frac a x + \frac {a \paren {\... | Real Area Hyperbolic Cosecant of x over a in Logarithm Form | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Cosecant_of_x_over_a_in_Logarithm_Form | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Cosecant_of_x_over_a_in_Logarithm_Form | [
"Inverse Hyperbolic Cosecant"
] | [] | [] |
proofwiki-9896 | Primitive of Root of x squared plus a squared/Logarithm Form | :$\ds \int \sqrt {x^2 + a^2} \rd x = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \map \ln {x + \sqrt {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \sqrt {x^2 + a^2} \rd x
| r = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \arsinh \frac x a + C
| c = Primitive of $\sqrt {x^2 + a^2}$ in $\arsinh$ form
}}
{{eqn | r = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \paren {\map \ln {x + \sqrt {x^2 + a^2} } - \ln a} + C
... | :$\ds \int \sqrt {x^2 + a^2} \rd x = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \map \ln {x + \sqrt {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \sqrt {x^2 + a^2} \rd x
| r = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \arsinh \frac x a + C
| c = [[Primitive of Root of x squared plus a squared/Inverse Hyperbolic Sine Form|Primitive of $\sqrt {x^2 + a^2}$ in $\arsinh$ form]]
}}
{{eqn | r = \frac {x \sqrt {x^2 + a^2} ... | Primitive of Root of x squared plus a squared/Logarithm Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared/Logarithm_Form | [
"Primitive of Root of x squared plus a squared"
] | [] | [
"Primitive of Root of x squared plus a squared/Inverse Hyperbolic Sine Form",
"Real Area Hyperbolic Sine of x over a in Logarithm Form",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-9897 | Primitive of Root of x squared plus a squared/Inverse Hyperbolic Sine Form | :$\ds \int \sqrt {x^2 + a^2} \rd x = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \sinh \theta
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \cosh \theta
| c = Derivative of Hyperbolic Sine
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = x
| r = a \sinh \theta
}}
{{eqn | ll= \leadsto
| l = x^2 ... | :$\ds \int \sqrt {x^2 + a^2} \rd x = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \sinh \theta
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \cosh \theta
| c = [[Derivative of Hyperbolic Sine]]
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = x
| r = a \sinh \theta
}}
{{eqn | ll= \leadsto
| l ... | Primitive of Root of x squared plus a squared/Inverse Hyperbolic Sine Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared/Inverse_Hyperbolic_Sine_Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared/Inverse_Hyperbolic_Sine_Form | [
"Primitive of Root of x squared plus a squared"
] | [] | [
"Derivative of Hyperbolic Sine",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Integration by Substitution",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9898 | Primitive of x squared over Root of x squared plus a squared/Inverse Hyperbolic Sine Form | :$\ds \int \frac {x^2 \rd x} {\sqrt {x^2 + a^2} } = \frac {x \sqrt {x^2 + a^2} } 2 - \frac {a^2} 2 \sinh^{-1} \frac x a + C$ | With a view to expressing the problem 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 = Power Rule for Derivatives
}}
{{end-eqn}}
and ... | :$\ds \int \frac {x^2 \rd x} {\sqrt {x^2 + a^2} } = \frac {x \sqrt {x^2 + a^2} } 2 - \frac {a^2} 2 \sinh^{-1} \frac x a + C$ | With a view to expressing the problem 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 = [[Power Rule for Derivatives]]
}}
{{end-eqn}}... | Primitive of x squared over Root of x squared plus a squared/Inverse Hyperbolic Sine Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_x_squared_plus_a_squared/Inverse_Hyperbolic_Sine_Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_x_squared_plus_a_squared/Inverse_Hyperbolic_Sine_Form | [
"Primitive of x squared over Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Primitive of x over Root of x squared plus a squared",
"Integration by Parts",
"Primitive of Root of x squared plus a squared/Inverse Hyperbolic Sine Form"
] |
proofwiki-9899 | Primitive of x squared over Root of x squared plus a squared/Logarithm Form | :$\ds \int \frac {x^2 \rd x} {\sqrt {x^2 + a^2} } = \frac {x \sqrt {x^2 + a^2} } 2 - \frac {a^2} 2 \map \ln {x + \sqrt {x^2 + a^2} } + C$ | With a view to expressing the problem 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 = Power Rule for Derivatives
}}
{{end-eqn}}
and ... | :$\ds \int \frac {x^2 \rd x} {\sqrt {x^2 + a^2} } = \frac {x \sqrt {x^2 + a^2} } 2 - \frac {a^2} 2 \map \ln {x + \sqrt {x^2 + a^2} } + C$ | With a view to expressing the problem 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 = [[Power Rule for Derivatives]]
}}
{{end-eqn}}... | Primitive of x squared over Root of x squared plus a squared/Logarithm Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_x_squared_plus_a_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_x_squared_plus_a_squared/Logarithm_Form | [
"Primitive of x squared over Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Primitive of x over Root of x squared plus a squared",
"Integration by Parts",
"Primitive of Root of x squared plus a squared/Logarithm Form"
] |
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