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Question
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
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Solution
Let I =`int ("x"/("x + 1")^2) "e"^"x"` dx
`= int "e"^"x" ((("x + 1") - 1)/("x + 1")^2)` dx
`= int "e"^"x"(("x + 1")/("x + 1")^2 - 1/("x + 1")^2)` dx
`= int "e"^"x" (1/("x + 1") - 1/("x + 1")^2)` dx
Put f(x) = `1/("x + 1")`
∴ f '(x) = `(-1)/("x + 1")^2`
∴ I = `int "e"^"x" ["f"("x") + "f" '("x")]` dx
`= "e"^"x" * "f"("x") + "c"`
∴ I = `"e"^"x" (1/("x + 1"))` + c
Notes
The answer in the textbook is incorrect.
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