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Question
Find `int e^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`.
Solution
Let I = `int e^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`
Take cot–1 x = t
∴ x = cot t
∴ dx = – cosec2 t dt
∴ I = `int e^t ((1 + cot^2 t - cot t)/(1 + cot^2 t))(-"cosec"^2t) dt`
= `int - e^t (("cosec"^2t - cot t))/("cosec"^2t) xx "cosec"^2 t dt`
= `int - e^t ("cosec"^2 t - cot t)dt`
= `int e^t (cot t - "cosec"^2t)dt`
∴ Now, taking f(t) = cot t
Then f'(t) = – cosec2 t
∴ I = `int e^t [f(t) + f^'(t)]dt`
= et f(t) + C
= `e^(cot^(–1)x) |cot (cot^-1 x)| + C`
= `e^(cot^(-1)x) xx x + C`
= `xe^(cot^(–1)x) + C`.
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