Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 3

Theorem

$\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$

Proof

\(\text {(1)}: \quad\)

\(\ds \int \sinh^n a x \cosh a x \rd x\)

\(=\)

\(\ds \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C\)

Primitive of $\sinh^n a x \cosh a x$

\(\ds \leadsto \ \ \)

\(\ds \int \sinh a x \cosh a x \rd x\)

\(=\)

\(\ds \frac {\sinh^2 a x} {2 a} + C\)

setting $n = 1$ in $(1)$

$\blacksquare$