Primitive of x over Power of Hyperbolic Sine of a x

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Theorem

$\ds \int \frac {x \rd x} {\sinh^n a x} = \frac {-x \cosh a x} {a \paren {n - 1} \sinh^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sinh^{n - 2} a x} - \frac {n - 2} {n - 1} \int \frac {x \rd x} {\sinh^{n - 2} a x}$


Proof

\(\ds \int \frac {x \rd x} {\sinh^n a x}\) \(=\) \(\ds \int x \csch^n a x \rd x\) Definition of Hyperbolic Cosecant
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \theta \csch^n \theta \rd \theta\) Substitution of $a x \to \theta$
\(\ds \leadsto \ \ \) \(\ds \frac 1 {a^2} \int \theta \csch^n \theta \rd \theta\) \(=\) \(\ds \frac 1 {a^2} \int \csch^2 \theta \times \theta \csch^{n - 2} \theta \rd \theta\) $\rd u = \csch^2 \theta$ and $v = \theta \csch^{n - 2} \theta$
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \paren {- \theta \coth \theta \csch^{n - 2} \theta - \int - \coth \theta \paren {- \paren {n - 2} \theta \coth \theta + 1} \csch^{n - 2} \theta \rd \theta}\) Integration by Parts
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \paren {- \theta \coth \theta \csch^{n - 2} \theta + \int \coth \theta \csch^{n - 2} \theta \rd \theta - \paren {n - 2} \int \theta \coth^2 \theta \csch^{n - 2} \theta \rd \theta}\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \paren {- \theta \coth \theta \csch^{n - 2} \theta + \int \coth \theta \csch^{n - 2} \theta \rd \theta - \paren {n - 2} \int \theta \paren {1 + \csch^2 \theta} \csch^{n - 2} \theta \rd \theta}\) Difference of Squares of Hyperbolic Cotangent and Cosecant
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \paren {- \theta \coth \theta \csch^{n - 2} \theta + \int \coth \theta \csch^{n - 2} \theta \rd \theta - \paren {n - 2} \int \theta \csch^{n - 2} \theta \rd \theta - \paren {n - 2} \int \theta \csch^n \theta \rd \theta}\) simplifying
\(\ds \leadsto \ \ \) \(\ds \frac {1 + \paren {n - 2} } {a^2} \int \theta \csch^n \theta \rd \theta\) \(=\) \(\ds \frac 1 {a^2} \paren {- \theta \coth \theta \csch^{n - 2} \theta + \int \coth \theta \csch^{n - 2} \theta \rd \theta - \paren {n - 2} \int \theta \csch^{n - 2} \theta \rd \theta}\) adding end term to both sides
\(\ds \leadsto \ \ \) \(\ds \frac 1 {a^2} \int \theta \csch^n \theta \rd \theta\) \(=\) \(\ds \frac 1 {a^2 \paren {n - 1} } \paren {- \theta \coth \theta \csch^{n - 2} \theta + \int \coth \theta \csch^{n - 2} \theta \rd \theta - \paren {n - 2} \int \theta \csch^{n - 2} \theta \rd \theta}\) rearranging for the intended primitive
\(\ds \) \(=\) \(\ds \frac 1 {a^2 \paren {n - 1} } \paren {- \theta \coth \theta \csch^{n - 2} \theta + \paren {- \frac {\csch^{n - 2} \theta } { \paren {n - 2} } } - \paren {n - 2} \int \theta \csch^{n - 2} \theta \rd \theta}\) Primitive of Power of Hyperbolic Cosecant of a x by Hyperbolic Cotangent of a x
\(\ds \) \(=\) \(\ds - \frac {\theta \cosh \theta} {a^2 \paren {n - 1} \sinh^{n - 1} \theta } - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sinh^{n - 2} \theta } - \frac {n - 2} {a^2 \paren {n - 1} } \int \frac {\theta} {\sinh^{n - 2} \theta} \rd \theta\) simplifying
\(\ds \leadsto \ \ \) \(\ds \int \frac {x \rd x} {\sinh^n a x}\) \(=\) \(\ds - \frac {a x \cosh a x} {a^2 \paren {n - 1} \sinh^{n - 1} a x } - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sinh^{n - 2} ax } - \frac {n - 2} {n - 1} \int \frac x {\sinh^{n - 2} a x} \rd x\) substituting back $\theta \to ax$
\(\ds \) \(=\) \(\ds \frac {- x \cosh a x} {a \paren {n - 1} \sinh^{n - 1} a x } - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sinh^{n - 2} ax } - \frac {n - 2} {n - 1} \int \frac {x \rd x} {\sinh^{n - 2} a x}\) simplifying

$\blacksquare$


Also see


Sources

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