Primitive of Power of x by Hyperbolic Sine of a x

From ProofWiki
Jump to navigation Jump to search


$\ds \int x^m \sinh a x \rd x = \frac {x^m \cosh a x} a - \frac m a \int x^{m - 1} \cosh a x \rd x + C$


With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$


\(\ds u\) \(=\) \(\ds x^m\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds m x^{m - 1}\) Derivative of Power

and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \sinh a x\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {\cosh a x} a\) Primitive of $\sinh a x$


\(\ds \int x^m \sinh a x \rd x\) \(=\) \(\ds x^m \paren {\frac {\cosh a x} a} - \int \paren {\frac {\cosh a x} a} \paren {m x^{m - 1} } \rd x + C\) Integration by Parts
\(\ds \) \(=\) \(\ds \frac {x^m \cosh a x} a - \frac m a \int x^{m - 1} \cosh a x \rd x + C\) Primitive of Constant Multiple of Function


Also see