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Question
Integrate the following with respect to x:
`"e"^(tan^-1 x) ((1 + x + x^2)/(1 + x^2))`
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Solution
I = `int "e"^(tan^-1 x) ((1 + x + x^2)/(1 + x^2)) "d"x`
I = `int "e"^(tan^-1 x) (1 + x + x^2) 1/(1 + x^2) "d"x`
Put tan–1 x = t
`1/(x1 + x^2) "d"x` = dt
x = tan t
∴ I = `int "e"^"t" [1 + tan "t" + tan^2"t"] "dt"`
= `int "e"^"t" [1 + tan^2"t" + tan "t"] "dt"`
= `int "e"^"t" [sec^2"t" + tan "t"] "dt"`
= `int "e"^"t" [tan "t" + sec^2"t"] "dt"`
f(x) = tan t
f'(x) = sec2t
`[int "e"^x ["f"(x) + "f"(x)] "d"x = "e"^x "f"(x) + "c"]`
∴ I = et tan t + c
I = `"e"^(tan^-1 (x)) * x + "c"`
I = `"e"^(tan^-1(x)) + "c"`
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