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Question
Integrate the following with respect to x.
`"e"^x [(x - 1)/(x + 1)^3]`
Solution
`"e"^x [(x - 1)/(x + 1)^3] = "e"^x [(x + 1 - 1 - 1)/(x + 1)^3]`
= `"e"^x [(x + 1)/(x + 1)^3 - 2/(x + 1)^3]`
= `"e"^x [1/(x + 1)^2 - 2/(x + 1)^3]`
Ler f(x) = `1/(x + 1)^2`
f'(x) = `(- 2)/(x + 1)^3`
So `int ("e"^x (x - 1))/(x + 1)^3 "d"x = int "e"^x ["f"(x) + "f'"(x)] "d"x`
= `"e"^x "f"(x) + "c"`
= `"e"^x/(x + 1)^2 + "c"`
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