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Question
Integrate the function:
`(2+ sin 2x)/(1+ cos 2x) e^x`
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Solution
Let `I = int (2 + sin 2x)/(1 + cos 2x) e^x dx`
`int (2 + 2 sin x cos x)/(1 + 2 cos^2 x - 1)e^x dx`
`= int (2 (1 + sin x cos x))/(2 cos^2 x) e^x dx`
`= int (sec^2 x * e^x + tan * e^x) dx`
`= int e^x (sec^2 x + tan x) dx`
Putting ex tan x = t
(ex sec2 x + tan x · ex)dx = dt
Hence, I = `int 1 * dt`
= t + C = ex tan x + C
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