Primitive of Square of Hyperbolic Sine Function

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int \sinh^2 x \rd x = \frac {\sinh 2 x} 4 - \frac x 2 + C$

where $C$ is an arbitrary constant.


Corollary

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

where $C$ is an arbitrary constant.


Proof

\(\ds \int \sinh^2 x \rd x\) \(=\) \(\ds \int \paren {\frac {\cosh 2 x - 1} 2} \rd x\) Square of Hyperbolic Sine
\(\ds \) \(=\) \(\ds \int \paren {\frac {\cosh 2 x} 2} \rd x - \int \frac 1 2 \rd x\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \int \paren {\frac {\cosh 2 x} 2} \rd x - \frac x 2 + C\) Primitive of Constant
\(\ds \) \(=\) \(\ds \frac 1 2 \paren {\frac {\sinh 2 x} 2} - \frac x 2 + C\) Primitive of Function of Constant Multiple and Primitive of Hyperbolic Cosine Function
\(\ds \) \(=\) \(\ds \frac {\sinh 2 x} 4 - \frac x 2 + C\) rearranging

$\blacksquare$


Sources