Question:
Evaluate the following integrals:
$\int \frac{1}{x \sqrt{x^{4}-1}} d x$
Solution:
Assume $x^{2}=t$
$2 x \cdot d x=d t$
$\Rightarrow \mathrm{dx}=\frac{\mathrm{dt}}{2 \mathrm{x}}$
Substituting $t$ and $d t$
$\Rightarrow \int \frac{\mathrm{dt}}{2 \mathrm{x}} \times \frac{1}{\mathrm{x} \times \sqrt{\mathrm{t}^{2}-1}}$
$\Rightarrow \int \frac{\mathrm{dt}}{2 \mathrm{x}^{2}} \times \frac{1}{\sqrt{\mathrm{t}^{2}-1}}$
$\Rightarrow \frac{1}{2} \int \frac{\mathrm{dt}}{\mathrm{t} \sqrt{\mathrm{t}^{2}-1}}$
$\Rightarrow \frac{1}{2} \mathrm{sec}^{-1} \mathrm{t}+\mathrm{c}$
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