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Question
Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
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Solution
Let I = `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
`= int "e"^"x" [((2 + "x") - 1)/(2 + "x")^2]` dx
`= int "e"^"x" [1/(2 + "x") - 1/(2 + "x")^2]`dx
Let f(x) = `1/(2 + "x")`
∴ f '(x) = `(-1)/(2 +"x")^2`
∴ I = `int "e"^"x" ["f"("x") + "f" '("x")]` dx
`= "e"^"x" * "f"("x") + "c"`
`= "e"^"x" * 1/(2 + "x")` + c
∴ I = `"e"^"x"/(2 + "x")` + c
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