Integrate solution: \int(\frac{\tan^{-1}x}{x-\tan^{-1}x})^2dx

Slade Higgins 2022-04-24 Answered
Integrate solution:
(tan1xxtan1x)2dx
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Answers (1)

louran20z47
Answered 2022-04-25 Author has 14 answers
With x=tan(u),dx=1cos2(u)du, thus
(tan1xxtan1x)2dx=(utan(u)u1cos(u))2du
Now, observe that
d(1sin(u)ucos(u))=usin(u)du(sin(u)ucos(u))2
This allows to integrate by parts:
=1x(1+(1+x2)arctan(x)xarctan(x))+C=1+xarctan(x)xarctan(x)+C
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