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Question

If I=cot1(a2ax+x2a2)dx, then I equals

A
xtan1(xa)(xa)tan1(xaa)+C
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B
a2log(2a22ax+x2)a2log(x2+a2)+C
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C
xtan1(xa)+(xa)tan1(xaa)+a2log(2a22ax+x2)+C
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D
none of these
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Solution

The correct option is D none of these
As a2ax+x2a2=1xa(axa) and cot1x=tan1(1x)
cot1(a2ax+x2a2)=tan1⎢ ⎢ ⎢ ⎢xa+(axa)1+xa(axa)⎥ ⎥ ⎥ ⎥=tan1(xa)+tan1(axa)
I=I1+I2
Where
I1=tan1(xa)dx=xtan1(xa)xax2+a2dx=xtan1(xa)a2log(x2+a2)
and
I2=tan1(axa)dx
Put axa=t
I2=atan1tdt=a[ttan1t12log(t2+1)]=(xa)tan1(axa)+a2log(2a22ax+x2)
Hence no option

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