Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 2

Theorem

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

Proof

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

\(=\)

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

\(\ds \)

\(=\)

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

Primitive of $\cosh^n a x \sinh a x$ using $n = 1$

\(\ds \)

\(=\)

\(\ds \frac {1 + \sinh^2 a x} {2 a} + C\)

Difference of Squares of Hyperbolic Cosine and Sine

\(\ds \)

\(=\)

\(\ds \frac 1 {2 a} + \frac {\sinh^2 a x} {2 a} + C\)

simplifying

\(\ds \)

\(=\)

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

subsuming $\dfrac 1 {2 a}$ into arbitrary constant

$\blacksquare$