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Question

Integrate the function tan1x

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Solution

Let I=1tan1xdx
Taking tan1 as first function and 1 as second function and integrating by parts, we obtain
I=tan1x1dx{(ddxtan1x)1dx}dx
=tan1xx11+x2xdx
=xtan1x122x1+x2dx
=xtan1x12log|1+x2|+C
=xtan1x12log(1+x2)+C

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