# The problem is the integral of &#x222B;<!-- ∫ --> <mrow class="MJX-TeXAtom-ORD">

The problem is the integral of
$\int \frac{-8x}{{x}^{4}-{a}^{4}}\phantom{\rule{thinmathspace}{0ex}}dx$
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humusen6p
Noting that
$I=\int \frac{-8x}{{x}^{4}-{a}^{4}}dx=-4\int \frac{d\left({x}^{2}\right)}{{\left({x}^{2}\right)}^{2}-{\left({a}^{2}\right)}^{2}}$
Let $y={x}^{2}$ and $b={a}^{2}$, then
$\begin{array}{rl}I& =-4\int \frac{dy}{{y}^{2}-{b}^{2}}\\ & =-4\int \frac{1}{2b}\left(\frac{1}{y-b}-\frac{1}{y+b}\right)dy\\ & =-\frac{2}{b}\left(\mathrm{ln}|y-b|-\mathrm{ln}|y+b|\right)+C\\ & =-\frac{2}{{a}^{2}}\mathrm{ln}|\frac{{x}^{2}-{a}^{2}}{{x}^{2}+{a}^{2}}|+C\\ & =\frac{2}{{a}^{2}}\mathrm{ln}|\frac{{x}^{2}+{a}^{2}}{{x}^{2}-{a}^{2}}|+C\end{array}$