Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

From ProofWiki
Jump to navigation Jump to search

Theorem

$\displaystyle \int \frac {\mathrm d x} {\sinh a x \cosh a x} = \frac 1 a \ln \left\vert{\tanh a x}\right\vert + C$


Proof

\(\displaystyle \int \frac {\mathrm d x} {\sinh a x \cosh a x}\) \(=\) \(\displaystyle \int \frac {\operatorname{sech} a x \ \mathrm d x} {\sinh a x}\) Definition of Hyperbolic Secant
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\operatorname{sech}^2 a x \ \mathrm d x} {\sinh a x \operatorname{sech} a x}\) multiplying top and bottom by $\operatorname{sech} a x$
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\operatorname{sech}^2 a x \ \mathrm d x} {\frac {\sinh a x} {\cosh a x} }\) Definition of Hyperbolic Secant
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\operatorname{sech}^2 a x \ \mathrm d x} {\tanh a x}\) Definition of Hyperbolic Tangent
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 a \ln \left\vert{\tanh a x}\right\vert + C\) Primitive of $\dfrac {\operatorname{sech}^2 a x} {\tanh a x}$

$\blacksquare$


Sources