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Question
Evaluate: `int_ (x + sin x)/(1 + cos x ) dx`
Solution
I = `int_ (x + sin x)/(1 + cos x ) dx`
= `int_ (x + sin x)/(2cos^2 x/2 dx`
= `∫x/(2cos^2 x/2)dx + ∫(2sin x/2 cos x/2)/(2 cos^2 x/2)dx`
= `(1)/(2) int_ xsec^2 x/2 "dx" + int_ tan x/2 dx`
= `(1)/(2) [ x int_ sec^2 x/2 "dx" - ( int_ (d)/(dx) (x) int_ sec^2 x/2 dx) dx] + int_ tan (x)/(2) dx`
= `(1)/(2) [ x .(tan x/2) /(1/2) - int_ (tan x/2 dx)/(1/2) ] + int_ tan (x)/(2) dx + c`
=` x tan x/2 - int_ tan x / 2 dx + int_ tan x /2 dx + c`
= `x tan x/2 + c`
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