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प्रश्न
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
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उत्तर
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
The answer in the textbook is incorrect.
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