Let, #10; #10; I=∫2( x^2+1 )^2dx #10; #10; I=2 ∫1( x^2+1 )^2dx #10; #10; #160; #10; #10;Applying reduction formula, we have #10; ...

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Integration by Substitution

Integrate: ∫...

Question

Integrate:
$\int \frac{2}{(x^{2} + 1)^{2}} d x$

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Solution

Let,

$I = \int \frac{2}{{(x^{2} + 1)}^{2}} d x$

$I = 2 \int \frac{1}{{(x^{2} + 1)}^{2}} d x$

Applying reduction formula, we have

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

Therefore,

$I = \frac{x}{(x^{2} + 1)} + \int \frac{1}{x^{2} + 1} d x + C$

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

Hence, this is the required result.

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