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Question
`int "e"^x ((1 + x^2))/(1 + x)^2 "d"x`
Solution
Let I = `int "e"^x ((1 + x^2))/(1 + x)^2 "d"x`
= `int "e"^x [(x^2 - 1 + 2)/(1 + x)^2] "d"x`
= `int "e"^x [(x^2- 1)/(x + 1)^2 + 2/(x+ 1)^2] "d"x`
= `int "e"^x [(x- )/(x + 1) + 2/(x + 1)^2] "d"x`
Put f(x) = `(x - 1)/(x + 1)`
∴ f'(x) = `((x +1)(1 - 0) - (x - 1)(1 + 0))/(x + 1)^2`
= `2/(x + 1)^2`
∴ I = `int "e"^x ["f"(x) + "f'"(x)] "d"x`
= ex.f(x) + c
= `"e"^x((x - 1)/(x + 1)) + "c"`
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