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Question
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
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Solution
Let I = `int e^x ((1 + sin x)/(1 + cos x)).dx`
= `int e^x [(1 + 2sin x/2 cos x /2)/(2 cos^2 x/2)].dx`
= `int e^x [(1)/(2cos^2 x/2) + (2sin x/2 cos x/2)/(2cos^2 x/2)].dx`
= `int e^x[1/2 sec^2 x/2 + tan (x/2)].dx`
Put f(x) = `tan (x/2)`
∴ f'(x) = `d/dx [tan x/2]`
= `sec^2 x/(2).(1)/(2)`
= `(1)/(2) sec^2 x/(2)`
∴ I = `int e^x [f(x) + f'(x)].dx`
= ex f(x) + c
= `e^x. tan (x/2) + c`.
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