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Question
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
Solution
Let I = `int e^x[x/(x + 1)^2].dx`
= `int e^x [((x + 1) - 1)/(x + 1)^2].dx`
= `int e^x [1/(x + 1) - 1/(x + 1)^2].dx`
Let f(x) = `(1)/(x + 1)`
= `(x + 1)^-1`
∴ f'(x) = `d/dx(x + 1)^-1`
= `-(x + 1)^-2 d/dx(x + 1)`
= `(-1)/(x + 1)^2 xx 1`
= `(-1)/(x + 1)^2`
∴ I = `int e^x [f(x) + f'(x)].dx`
= ex.f(x) + c
= `e^x/(x + 1) + c`.
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