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Question

Question: \[\int_{0}^{\infty}\frac{xdx}{(1 + x)(1 + x^{2})} =\]...

∫0∞xdx(1+x)(1+x2)=\int_{0}^{\infty}\frac{xdx}{(1 + x)(1 + x^{2})} =

A

π4\frac{\pi}{4}

B

π3\frac{\pi}{3}

C

π6\frac{\pi}{6}

D

None of these

Answer

π4\frac{\pi}{4}

Explanation

Solution

I=∫0∞xdx(1+x)(1+x2)I = \int_{0}^{\infty}\frac{xdx}{(1 + x)(1 + x^{2})}

Put x=tan⁡θx = \tan\theta, we get

I=∫0π/2tan⁡θ1+tan⁡θdθ=∫0π/2sin⁡θcos⁡θ+sin⁡θdθ=π4I = \int_{0}^{\pi/2}{\frac{\tan\theta}{1 + \tan\theta}d\theta = \int_{0}^{\pi/2}{\frac{\sin\theta}{\cos\theta + \sin\theta}d\theta = \frac{\pi}{4}}}.