Evaluate: `int xtan^(-1)x dx`

Explanation

Let `I=int x tan ^{-1} x d x`

Taking tan ^{-1} x as first function and x as the second function and integrating by parts, we obtain


`I=tan ^{-1} x int x d x-int{(frac{d}{d x} tan ^{-1} x) int x d x} d x `
`
=tan ^{-1} x(frac{x^{2}}{2})-int frac{1}{1+x^{2}} cdot frac{x^{2}}{2} d x `
`
=frac{x^{2} tan ^{-1} x}{2}-frac{1}{2} int frac{x^{2}}{1+x^{2}} d x `
`
=frac{x^{2} tan ^{-1} x}{2}-frac{1}{2} int(frac{x^{2}+1}{1+x^{2}}-frac{1}{1+x^{2}}) d x`
`
=frac{x^{2} tan ^{-1} x}{2}-frac{1}{2} int(1-frac{1}{1+x^{2}}) d x `
`
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