Primitive of Reciprocal of Square of Hyperbolic Cosine of a x

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Theorem

$\ds \int \frac {\d x} {\cosh^2 a x} = \frac {\tanh a x} a + C$


Proof

\(\ds \int \frac {\d x} {\cosh^2 a x}\) \(=\) \(\ds \int \sech^2 a x \rd x\) Definition 2 of Hyperbolic Secant
\(\ds \) \(=\) \(\ds \frac {\tanh a x} a + C\) Primitive of $\sech^2 a x$

$\blacksquare$


Also see


Sources