Primitives of Hyperbolic Functions
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Theorem
This page gathers together primitives of hyperbolic functions.
In the below, $C$ is an arbitrary constant throughout.
Primitive of Hyperbolic Sine Function
- $\ds \int \sinh x \rd x = \cosh x + C$
Primitive of Hyperbolic Cosine Function
- $\ds \int \cosh x \rd x = \sinh x + C$
Primitive of Hyperbolic Tangent Function
- $\ds \int \tanh x \rd x = \map \ln {\cosh x} + C$
Primitive of Hyperbolic Cotangent Function
- $\ds \int \coth x \rd x = \ln \size {\sinh x} + C$
where $\sinh x \ne 0$.
Hyperbolic Secant Function: Arcsine Form
- $\ds \int \sech x \rd x = \map \arcsin {\tanh x} + C$
Hyperbolic Secant Function: Arctangent of Exponential Form
- $\ds \int \sech x \rd x = 2 \map \arctan {e^x} + C$
Primitive of Hyperbolic Cosecant Function: Logarithm Form
- $\ds \int \csch x \rd x = -\ln \size {\csch x + \coth x} + C$
where $\csch x + \coth x \ne 0$.
Primitive of Hyperbolic Cosecant Function: Hyperbolic Tangent Form
- $\ds \int \csch x \rd x = \ln \size {\tanh \frac x 2} + C$
where $\tanh \dfrac x 2 \ne 0$.
Primitive of Hyperbolic Cosecant Function: Inverse Hyperbolic Cotangent Form
- $\ds \int \csch x \rd x = -2 \map {\coth^{-1} } {e^x} + C$
Primitive of Hyperbolic Cosecant Function: Inverse Hyperbolic Cotangent of Hyperbolic Cosine Form
- $\ds \int \csch x \rd x = -\map {\coth^{-1} } {\cosh x} + C$