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Question
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
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Solution
Let I = `int (x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x).dx`
= `int a^(x + tan^-1x).((x^2 + 2)/(x^2 + 1)).dx`
Put x + tan–1x = t
∴ `(1 + 1/(1 + x^2)).dx` = dt
∴ `((1 + x^2 + 1)/(1 + x^2)).dx` = dt
∴ `((x^2 + 2)/(x^2 + 1)).dx` = dt
∴ I = `int a^t dt = a^t/loga + c`
= `(a^(x + tan^-1 x))/loga + c`.
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