- 0 -x dx-1-x1-x2 is equal to

The correct option is A π/4 Let I=∫ _ 0 ^∞ nbsp; xdx ( 1+x ) ( 1+ x ^ 2 ) nbsp; nbsp; Using partial fraction =∫ _ 0 ^∞ nbsp;( x+1 2( 1 ...

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Byju's Answer

Standard XII

Mathematics

Differentiation Using Substitution

∫ 0 ∞x dx/1+x...

Question

$\int_{0}^{\infty} \frac{x d x}{(1 + x) (1 + x^{2})}$ is equal to

\frac{π}{4}

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\frac{π}{2}

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π

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none of these

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Solution

The correct option is A $\frac{π}{4}$
Let $I = \int_{0}^{\infty} \frac{x d x}{(1 + x) (1 + x^{2})}$
Using partial fraction
$= \int_{0}^{\infty} (\frac{x + 1}{2 (1 + x^{2})} - \frac{1}{2 (1 + x)}) d x = {({lim}_{b \to \infty} \frac{1}{2} log (1 + x^{2}) - \frac{1}{2} log (1 + x) + \frac{1}{2} {t a n}^{- 1} x)}_{0}^{b} = {lim}_{b \to \infty} (\frac{1}{2} log (1 + b) - \frac{1}{2} log (1 + b) + \frac{1}{2} {t a n}^{- 1} b) = \frac{π}{4}$

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