Sum

**Evaluate the following.**

`int "e"^"x" "x - 1"/("x + 1")^3` dx

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#### Solution

Let I = `int "e"^"x" (("x - 1"))/("x + 1")^3 "dx" = int "e"^"x" (("x" + 1 - 1 - 1))/("x + 1")^3`dx

`= int "e"^"x" [("x + 1")/(("x + 1")^3) - 2/("x + 1")^3]` dx

`= int "e"^"x" [1/("x + 1")^2 - 2/("x + 1")^3]` dx

Put f(x) = `1/("x + 1")^2`

∴ f '(x) = `(-2)/("x + 1")^3`

∴ I = `int "e"^"x" ["f"("x") + "f" '("x")]` dx

`= "e"^"x" "f"("x") + "c"`

`= "e"^"x" xx 1/("x + 1")^2 + "c"`

∴ I = `"e"^"x"/("x + 1")^2 + "c"`

#### Notes

[**Note:** Answer in the textbook is incorrect.]

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