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Question

Evaluate (x1)(x+1)x3+x2+xdx.

A
I=2tan1x2+x+1x+C
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B
I=2tan1x2+2x+1x+C
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C
I=2sec1x2+x+1x+C
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D
I=2sec1x2+2x+1x+C
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Solution

The correct option is A I=2tan1x2+x+1x+C
Let I=(x1)(x+1)x3+x2+xdx=(x21)(x+1)2x3+x2+xdx
=x2(11x2)(x2+2x+1)x3+x2+xdx
=x2(11x2)x(x+2+1x).xx+1+1xdx
Put x+1x=t, (11x2)dx=dt
=dt(t+2)t+1, which reduces to dxPQ.
Put t+1=z2
I=2zdz(z2+1)z2=2dzz2+1=2tan1(z)+C
=2tan1(t+1)+C=2tan1x2+x+1x+C

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