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Evaluate the following. ∫exx - 1(x + 1)3 dx - Mathematics and Statistics

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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