Question

Evaluate the following.

`int "e"^"x" "x"/("x + 1")^2` dx

Answer 1

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