Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \frac{x^{2}}{x^{6}-a^{6}} d x$

Solution:

let $I=\int \frac{x^{2}}{x^{6}-a^{6}} d x$

$=\int \frac{x^{2}}{\left(x^{3}\right)^{2}-\left(a^{3}\right)^{2}} d x$

Let $\mathrm{x}^{3}=\mathrm{t} \ldots . .$ (i)

$\Rightarrow 3 x^{2} d x=d t$

$I=\frac{1}{3} \int \frac{1}{t^{2}-\left(a^{3}\right)^{2}} d t$

$I=\frac{1}{3} \times \frac{1}{2 \times a^{3}} \log \left|\frac{t-a^{3}}{t+a^{3}}\right|+c$

[since, $\left.\int \frac{1}{x^{2}-(a)^{2}} d x=\frac{1}{2 \times a} \log \left|\frac{x-a}{x+a}\right|+c\right]$

$I=\frac{1}{6 a^{3}} \log \left|\frac{x^{3}-a^{3}}{x^{3}+a^{3}}\right|+c[u \operatorname{sing}(i)]$

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