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Question
`int "e"^x x/(x + 1)^2 "d"x`
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Solution
Let I = `int "e"^x(x/((x + 1)^2))"d"x`
= `int"e"^x (((x + 1) - 1)/(x + 1)^2)"d"x`
= `int"e"^x ((x + 1)/(x + 1)^2 - 1/(x + 1)^2)"d"x`
= `int"e"^x (1/(x + 1) - 1/(x + 1)^2)"d"x`
Put f(x) = `1/(x + 1)`
∴ f'(x) = `(-1)/(x + 1)^2`
∴ I = `int"e"^x["f"(x) + "f'"(x)]"d"x`
= `"e"^x*"f"(x) + "c"`
∴ I = `"e"^x (1/(x + 1)) + "c"`
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