Primitive of x by Square of Hyperbolic Sine of a x

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Theorem

$\displaystyle \int x \sinh^2 a x \rd x = \dfrac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} - \frac {x^2} 4 + C$


Proof

With a view to expressing the primitive in the form:

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

let:

\(\displaystyle u\) \(=\) \(\displaystyle x\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac {\d u} {\d x}\) \(=\) \(\displaystyle 1\) Derivative of Identity Function


and let:

\(\displaystyle \frac {\d v} {\d x}\) \(=\) \(\displaystyle \sinh^2 a x\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle \frac {\sinh 2 a x} {4 a} - \frac x 2\) Primitive of $\sinh^2 a x$


Then:

\(\displaystyle \int x \sinh^2 a x \rd x\) \(=\) \(\displaystyle x \paren {\frac {\sinh 2 a x} {4 a} - \frac x 2} - \int \paren {\frac {\sinh 2 a x} {4 a} - \frac x 2} \times 1 \rd x + C\) Integration by Parts
\(\displaystyle \) \(=\) \(\displaystyle \frac {x \sinh 2 a x} {4 a} - \frac {x^2} 2 - \frac 1 {4 a} \int \sinh 2 a x \rd x + \frac 1 2 \int x \rd x + C\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \frac {x \sinh 2 a x} {4 a} - \frac {x^2} 2 - \frac 1 {4 a} \int \sinh 2 a x \rd x + \frac {x^2} 4 + C\) Primitive of Power
\(\displaystyle \) \(=\) \(\displaystyle \frac {x \sinh 2 a x} {4 a} - \frac {x^2} 2 - \frac 1 {4 a} \frac {\cosh 2 a x} {2 a} + \frac {x^2} 4 + C\) Primitive of $\sinh a x$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} - \frac {x^2} 4 + C\) simplifying

$\blacksquare$


Also see


Sources