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Question

Integrate the function.
xtan1xdx.

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Solution

Let I=xtan1xdx
On taking tan1x as first function and x as second function and integrating by parts, we get
( Inverse functions comes before algebraic function in ILATE)
I=tan1xxdx[ddx(tan1x)xdx]dx
=tan1x.x22[11+x2.x22]dx=x22.tan1x12[x2+111+x2]dx
[Add and subtract 1 in numerator of second term]
=x22.tan1x12[1+x21+x2dx11+x2dx]=x22tan1x12[1dx11+x2dx]=x22tan1x12x+12tan1x+C=(x2+12)tan1x12x+C


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