First slide

Methods of integration

Question

$I = \int \frac{x^{2} - 1}{x^{3} \sqrt{2 x^{4} - 2 x^{2} + 1}} d x$ equal to

Moderate

A

$\sqrt{2 x^{4} - 2 x^{2} + 1} + C$
B

$\frac{\sqrt{2 x^{4} - 2 x^{2} + 1}}{x^{3}} + C$
C

$\frac{\sqrt{2 x^{4} - 2 x^{2} + 1}}{x} + C$
D

$\frac{\sqrt{2 x^{4} - 2 x^{2} + 1}}{2 x^{2}} + C$

Solution

$I = \int \frac{((1 / x^{3}) - (1 / x^{5}))}{\sqrt{2 - (2 / x^{2}) + (1 / x^{4})}} d x$
put $2 - \frac{2}{x^{2}} + \frac{1}{x^{4}} = t^{2} \Rightarrow (\frac{4}{x^{3}} + \frac{4}{x^{5}}) d x = 2 t d t$
$∴$ $\begin{aligned} I = \frac{1}{2} \int \frac{t d t}{t} = \frac{1}{2} t + C \\ = \frac{1}{2} \sqrt{2 - \frac{2}{x^{2}} + \frac{1}{x^{4}}} + C \\ = \frac{\sqrt{2 x^{4} - 2 x^{2} + 1}}{2 x^{2}} + C \end{aligned}$

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