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प्रश्न
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate:
∫ x tan-1 x dx
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उत्तर
Let I = `int x tan^-1 x . dx`
= `int (tan^-1 x)x . dx`
= `(tan^-1 x) int x . dx - int[{d/dx(tan^-1 x) intx.dx}] . dx`
= `(tan^-1x) (x^2/2) - int (1/(1 + x^2)) (x^2/2) . dx`
= `(x^2 tan^-1)/(2) - (1)/(2) int x^2/(x^2 + 1) . dx`
= `x^2/(2) tan^-1x - (1)/(2) ((x^2 + 1)-1)/(x^2 + 1) ⋅ dx`
= `x^2/(2)tan^-1x - (1)/(2)[int(1 - 1/(x^2 + 1)) ⋅ dx]`
= `x^2/(2)tan^-1x - (1)/(2)[int 1 . dx - int(1)/(x^2 + 1) . dx]`
= `x^2/(2)tan^-1 x - (1)/(2)(x - tan^-1x) + c`.
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