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Question
Evaluate the following:
`int "e"^(tan^-1x) ((1 + x + x^2)/(1 + x^2)) "d"x`
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Solution
Let I = `int "e"^(tan^-1x) ((1 + x + x^2)/(1 + x^2)) "d"x`
Put `tan^-1x` = t
⇒ `1/(1 + x^2) * "d"x` = dt
= `int "e"^"t" (1 + tan "t" + tan^2 "t")"dt"`
= `int "e"^"t" (sec^2 "t" + tan "t")"dt"`
Here f(t) = tan t
∴ f'(t) = sec2t
= `"e"^"t" * "f"("t")`
= `"e"^"t" tan "t"`
= `"e"^(tan^-1x) * x + "c"` ....`[because int "e"^x ["f"(x) + "f'"(x)]"d"x = "e"^2"f"(x) + "C"]`
Hence, I = `"e"^(tan^-1x) * x + "C"`.
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