First slide

Methods of integration

Question

$\int \frac{dx}{(1 + x^{2}) \sqrt{1 - x^{2}}}$ is equal to

Moderate

A

$- \frac{1}{\sqrt{2}} \tan^{- 1} (\frac{\sqrt{1 - x^{2}}}{\sqrt{2} x}) + C$
B

$\frac{1}{\sqrt{2}} \tan^{- 1} (\frac{\sqrt{1 - x^{2}}}{\sqrt{2} x}) + C$
C

$- \tan^{- 1} (\frac{\sqrt{1 - x^{2}}}{\sqrt{2} x}) + C$
D

None of these

Solution

Put $x = \frac{1}{t} \Rightarrow dx = \frac{- 1}{t^{2}} dt$
$\begin{aligned} ∴ I = \int \frac{- dt}{t^{2} (1 + \frac{1}{t^{2}}) \sqrt{1 - {(\frac{1}{t})}^{2}}} \\ = - \int \frac{tdt}{(t^{2} + 1) \sqrt{t^{2} - 1}} \end{aligned}$
Again, put $t^{2} - 1 = y^{2} \Rightarrow tdt = ydy$
$\begin{array}{l} ∴ l = - \int \frac{ydy}{(y^{2} + 2) y} \\ = - \frac{1}{\sqrt{2}} \tan^{- 1} (\frac{y}{\sqrt{2}}) + C \\ = - \frac{1}{\sqrt{2}} \tan^{- 1} (\frac{\sqrt{1 - x^{2}}}{\sqrt{2} x}) + C \end{array}$

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