Evaluate the following: dx x x^4 -1 Hint: Put x^2= - Vidyadip

Question

Evaluate the following:
$\int\frac{\text{dx}}{\text{x}\sqrt{\text{x}^4}-1}$
Hint: Put $\text{x}^2=\sec\theta$

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Answer

Let $\text{I}=\int\frac{\text{dx}}{\text{x}\sqrt{\text{x}^4}-1}$
Put $\text{x}^2=\sec\theta\Rightarrow\theta=\sec^{-1}\text{x}^2$
$\Rightarrow 2\text{xdx} =\sec\theta.\tan\theta \text{d}\theta$
$\therefore\ \text{I}=\frac{1}{2}\int\frac{\sec\theta\cdot\tan\theta}{\sec\theta\cdot\tan\theta}\text{d}\theta$ $=\frac{1}{2}\int\text{d}\theta=\frac{1}{2}\theta+\text{C}$
$=\frac{1}{2}\sec^{-1}(\text{x}^2)+\text{C}$

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