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Question
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
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Solution
Let `I = int (e^x(1 + sin x))/(1 + cos x) dx`
`I = int e^x [(1 + sin x/2 cos x/2)/(2 cos^2 x/2)] dx`
`= inte^x [1/2 sec^2 x/2 + tan x/2] dx`
`I = inte^x [tan x/2 + 1/2 sec^2 x/2] dx`
`= e^x {tan x/2 + [d/dx(tan x/2)]}`
`= e^x * tan (x/2) + C` `...[∵ int e^x (f(x) + f' (x)) dx = e^x f (x) + C]`
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