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Question
Integrate the following with respect to x:
`"e"^x ((2 + sin 2x)/(1 + cos 2x))`
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Solution
I = `int "e"^x ((2 + sin 2x)/(1 + cos 2x)) "d"x`
I = `int "e"^x ((2 + 2 sin x cos x)/(2 cos^2 x)) "d"x`
I = `int "e"^x ((2(1 + sin x cos x))/(2 cos^2x)) "d"x`
I = `int "e"^x ((1 + sin x cos x)/(cos^2x)) "d"x`
I = `int "e"^x [1/(cos^2x) + (sin x cos x)/(cos^2x)] "d"x`
I = `int "e"^x [sec^2x + sinx/cosx] "d"x`
I = `int "e"^x [sec^2x + tan x] "d"x`
I = `int "e"^x [tan x + sec^2x] "d"x`
Take f(x) = tan x
f'(x) = sec2 x
`[int "e"^x ["f"(x) + "f"(x)] "d"x = "e"^x "f"(x) + "c"]`
I = ex tan x + c
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