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Question

Evaluate dx(x2+a2)3.

A
I=x4a2(x2+a2)2+34a2{x2a2(x2+a2)+12a3tan1(xa)}+C
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B
I=x2a2(x2+a2)2+34a2{xa2(x2+a2)+1a3tan1(xa)}+C
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C
I=x4a2(x2+a2)2+14a2{x2a2(x2+a2)+12a3sec1(xa)}+C
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D
I=xa2(x2+a2)2+32a2{x2a2(x2+a2)+12a3tan1(xa)}+C
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Solution

The correct option is A I=x4a2(x2+a2)2+34a2{x2a2(x2+a2)+12a3tan1(xa)}+C
Let I=dx(x2+a2)3 ....(i)
and I1=1(x2+a2)2dx ......(ii)
I1=1(x2+a2)2.1dx
=1(x2+a2)2.x2(2x)(x2+a2)3xdx
=x(x2+a2)2+4x2+a2a2(x2+a2)3dx
=x(x2+a2)2+41(x2+a2)2dx4a2dx(x2+a2)3
I1=x(x2+a2)2+4I14a2.I [using equations (i) and (ii)]
4a2.I=x(x2+a2)2+3I1
I=x4a2(x2+a2)2+34a2I1 (iii)
I=x4a2(x2+a2)2+34a2{x2a2(x2+a2)+12a3tan1(xa)}+C

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