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Question

Integrate the function.
xsin1xdx

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Solution

Let I=xsin1xdx
On taking sin1x as first function and x as second function and integrating by parts, we get
( Inverse functions comes before, algebraic functions in ILATE)
I=sin1xxdx[ddx(sin1x)xdx]dx=sin1x.x22[11x2.x22]dx=x22.sin1x+[11x21x2.12]dx
(Add and subtract 1 in numerator of second term)
=x22.sin1x1211x2dx+121x21x2dx=x22.sin1x12sin1x+121x2dxI=x22.sin1x12sin1x+12[12x1x2+12sin1x]+CI=sin1x[x2214]+14x1x2+CI=sin14(2x21)+14x1x2+C


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