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Question
`int (x + sinx)/(1 - cosx) "d"x`
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Solution
Let I = `int (x + sinx)/(1 - cosx) "d"x`
= `int ((x + 2 sin x/2 cos x/2)/(2 sin^2 x/2)) "d"x`
= `int (x/(2sin^2 x/2) + (2sin x/2 cos x/2)/(2sin^2 x/2)) "d"x`
= `1/2 int x "cosec"^2 x/2 "d"x + int (cos x/2)/(sin x/2) "d"x`
= `1/2[x int "cosec"^2 x/2 "d"x - int ("d"/("d"x)(x) int "cosec"^2 x/2 "d"x) "d"x] + int cot x/2 "d"x`
= `1/2[x((-cot x/2)/(1/2)) - int1 * ((- cot x/2)/(1/2)) "d"x] + int cot x/2 "d"x`
= `1/2(-2x cot x/2 + 2 int cot x/2 "d"x) + int cot x/2 "d"x`
= `- x cot x/2 + int cot x/2 "d"x + int cot x/2 "d"x`
= `- x cot x/2 + 2 int cot x/2 "d"x`
= `- x cot x/2 + 2 * (log|sin(x/2)|)/(1/2) + "c"`
= `- x cot x/2 + 4log |sin(x/2)| + "c"`
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