Evaluate `int_(0)^(pi)(xdx)/(1+cosalphasinx),0ltalphaltpi`.

Answer

`(alphapi)/(sin alpha)`

Explanation

Let `I= int _(0)^(pi) (x)/(1+ cos alpha sin x)dx` . . . (i)
`rArr I = int _(0)^(pi) ((pi-x))/(1+cos alpha sin (pi- x))dx`
` rArr I= int _(0)^(pi) ((pi-x))/(1+ cos alpha sin a )dx` . . . (ii)
On adding Eqs . (i) and (ii) , we get
`rArr 2 I = pi int _(0)^(pi) ("sec "^(2)(x)/(2)dx)/((1+ " tan" ^(2)(x)/(2))+2 cos alpha " tan " (x)/(2))`
Put `"tan " (x)/ (2)= t rArr "sec" ^(2)(x)/(2) dx = 2 dt`
`rArr 2 I = 2 pi int _(0)^(oo) (dt)/((t + cos alpha )^(2)+sin ^(2)alpha)`
`I= (pi)/(sinalpha) [ tan ^(-1) ((t- coas alpha)/( sin alpha))]_(0)^(oo)`
` = (pi)/(sin alpha)[ tan^(-1)(oo)- tan ^(-1) (cot alpha0]`
` = (pi)/(sin alpha)((pi)/(2)-((pi)/(2)- alpha))=(alphapi)/( sinalpha)`
`:. I= (alpha pi)/( sin alpha)`