Primitive of Square of Hyperbolic Secant of a x over Hyperbolic Tangent of a x

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int \frac {\sech^2 a x \rd x} {\tanh a x} = \frac 1 a \ln \size {\tanh a x} + C$


Proof

\(\ds \frac {\d} {\d x} \tanh a x\) \(=\) \(\ds a \sech^2 a x\) Derivative of $\tanh a x$
\(\ds \leadsto \ \ \) \(\ds \int \frac {a \sech^2 a x \rd x} {\tanh a x}\) \(=\) \(\ds \ln \size {\tanh a x} + C\) Primitive of Function under its Derivative
\(\ds \leadsto \ \ \) \(\ds \int \frac {\sech^2 a x \rd x} {\tanh a x}\) \(=\) \(\ds \frac 1 a \ln \size {\tanh a x} + C\) Primitive of Constant Multiple of Function

$\blacksquare$


Also see


Sources