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Question

Evaluate :- etan1x.1+x+x21+x2dx.

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Solution

I=etan1x(1+x+x21+x2)dx

Let tan1x=tx=tant,x2=tan2t11+x2dx=dtI=et1+tant+tan2t1+x2(1+x2)dt=et(1+tant+tan2t)dt=et(tant+sec2t)dt{ex(f(x)+f(x))=exf(x)+C}=et(tant)+C=xetan1x+C

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