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Question
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
Solution
Let I = `int ((x - 1)^2)/(x^2 + 1)^2.dx`
= `int (x^2 - 2x + 1)/(x^2 + 1)^2.dx`
= `int ((x^2 + 1) - 2x)/(x^2 + 1)^2.dx`
= `int [(x^2 + 1)/(x^2 + 1)^2 - (2x)/(x^2 + 1)^2].dx`
= `int (1)/(x^2 + 1)dx - int (2x)/(x^2 + 1)^2.dx`
= I1 – I2 ...(Let)
In I2, Put x2 + 1 = t
∴ 2x dx = dt
= I = `int (1)/(x^2 + 1).dx - int t^-2 dt`
= `tan^-1 x - t^-1/((-1)) + c`
= `tan^-1 x + (1)/(x^2 + 1) + c`.
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