Question

Find:

`int(x sin x)/(1 + cos x)  dx`

Answer 1

I = `int(x sin x)/(1 + cos x)  dx`

I = `int((x + 2sin x//2 cos x//2))/(2 cos^2 x//2)  dx`

= `int x/(2 cos^2  x/2)  dx + int (2 sin  x/2 cos  x/2)/(2 cos^2  x/2)`

= `1/2intx sec^2  x/2  dx + int tan  x/2  dx`

Intrigration by parts,

= `1/2[x int sec^2  x/2  dx - int (d/dx  x int sec^2x/2)dx + int tan  x/2  dx]`

= `1/2[x ⋅ 2tan  x/2 - int 1 ⋅ 2 tan  x/2  dx + tan  x/2  dx]`

= `x tan  x/2 - int tan  x/2  dx + int tan  x/2  dx`

= `x tan  x/2 + C`