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Question

The value of the integral 0 x dx(1+x)(1+x2) is equal to

A
π
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B
π4
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C
π2
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D
2π
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Solution

The correct option is B π4
We have,
Let I=0 x dx(1+x)(1+x2)Let x=tan θdx=sec2 θ dθWhen x=,θ=π2 and x=0,θ=0 I=π/20 tan θ sec2 θ (1+tan θ)(1+tan2 θ)dθI=π/20 sin θcos θ sec2 θ (1+sin θcos θ)sec2 θdθI=π/20 sin θsin θ+cos θdθ ....(1]I=π/20 sin (π2θ)sin (π2θ)+cos (π2θ)dθI=π/20 cos θsin θ+cos θdθ ....(2]Adding (1] and (2], we get2 I=π/20 sin θ+cos θsin θ+cos θdθ2 I=π/20 1dθ2 I=(x]π/20=π20I=π4

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