Question

Find the integral. \(\int x \operatorname{csch}^{2} \frac{x^{2}}{2} d x\)

Short answer

The result of evaluating the integral will be the final result of the integral after implementing the u-substitution, breaking down the integral, and integrating each part separately.

Step-by-step solution

Identifying a suitable substitution

First, let's choose a suitable substitution. In this case, by identifying the hyperbolic cosecant function, we notice that the argument of this function can be a potential choice for substitution. Hence, let \(u = x^2/2\). Taking derivative on both sides will give us \(du = x \, dx\).

Substitute Variable

Once the substitution is determined, the next step is to substitute back into the integral. The integral becomes: \(\int \operatorname{csch}^{2} u \, du\). The function \( \operatorname{csch} u = 1/ \sinh u\) and we know \( \sinh^2 u = \cosh^2 u - 1 \), so \( 1/ \sinh^2 u = \cosh^2 u - 1\). Therefore, replace csch2 with this alternate form.

Solve the Integral

The integral now becomes: \(\int (\cosh^2 u - 1) \, du\). Solving this by splitting it into two separate integrals: \(\int \cosh^2 u \, du - \int du\). Use the power-reduction identity \(\cosh^2 u = (1 + \cosh 2u)/2\) to simplify the integral. After solving the individual integrals, replace \(u\) with \(x^2/2\) to get the final answer.