Evaluate the following integrals:
Question:

$\int \frac{x^{2}-2}{x^{5}-x} d x$

Solution:

By partial fractions,

$=\frac{x^{2}-2}{x^{2}-5}=\frac{x^{2}-2}{(x-1) x(x+1)\left(x^{2}+1\right)}=\frac{A}{x-1}+\frac{B}{x}+\frac{C}{x+1}+\frac{D}{x^{2}+1}$

$=\int-\frac{d x}{4(x-1)}+\int \frac{2}{x} d x-\frac{1}{4} \int \frac{d x}{x+1}-\frac{3}{2} \int \frac{x d x}{x^{2}+1}$

$=-\frac{1}{4} \log (x-1)+2 \log x-\frac{1}{4} \log (x+1)-\frac{3}{4} \log \left(x^{2}+1\right)+c$

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