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Question
Integrate the function in x tan-1 x.
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Solution
Let `I = int x tan^-1 x dx`
`= tan^-1 x int x dx - int [(d/dx(tan^-1 x)) int (x dx)] dx`
`= tan^-1 x (x^2/2) - int 1/ (1 + x^2) * x^2/2 dx`
`= x^2/2 tan^-1 x - 1/2 int x^2/ (x^2 + 1) dx`
`= x^2/2 tan^-1 x - 1/2 int (x^2 + 1 - 1)/ (1 + x^2) dx`
`= x^2/2 tan^-1 x - 1/2 int (1 - 1/(1 + x^2)) dx`
`= x^2/2 tan^-1 x - 1/2 (x - tan^-1 x) + C`
`= x^2/2 tan^-1 x - 1/2 x + 1/2 tan^-1 x + C`
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