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Question

Let f(x)=x(1+x)2dx ;(x0). Then f(3)f(1) is equal to

A
π12+12+34
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B
π6+1234
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C
π6+12+34
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D
π12+1234
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Solution

The correct option is D π12+1234
Given:
f(x)=x(1+x)2dx
Put x=tan2θ
dx=2tanθsec2θdθ
f(θ)=2tan2θ.sec2θsec4θd θ
f(θ)=2sin2θ d θ
f(θ)=θsin 2θ2+C
f(θ)=θ12×2tanθ1+tan2θ+C
f(θ)=θtanθ1+tan2θ+C
f(x)=tan1xx1+x+C
Now,
f(3)f(1)=tan1(3)31+3tan1(1)+12
f(3)f(1)=π12+1234

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