Integrate the functions`1/(sqrt(sin^3xsin(x+alpha)))`

Explanation

`" Let "int=(1)/(sqrt(sin^(3)x sin (x+alpha)))dx`
`I= int (1)/(sqrt(sin^(3)x(sin x cos alpha + cos x sin alpha)))dx`
`=int(1)/(sqrt(sin^(4)x(cos alpha+cot x sin alpha)))dx`
`=int (1)/(sin^(2)xsqrt(cos alpha +cot xsin alpha))dx`
`=int ("cosec"^(2)x)/(sqrt(cos alpha+cot xsin alpha))dx`
`underset(rArr" ""cosec"^(2)x dx= -(dt)/(sinalpha))underset(rArr -"cosec"^(2) x sin alpha dx=dt)(" Let " cos alpha +cot x sin alpha=t)`
` :. I=(-1)/(sinalpha) int(dt)/(sqrt(t)) =(-1)/(sinalpha) (2sqrt(t)) +C`
`=(-1)/(sinalpha) 2sqrt(cosalpha +cot x sin alpha)+C`
`=(-2)/(sinalpha) sqrt(cosalpha+.(cos x sin alpha)/(sinx))+C`
`=(-2)/(sinalpha)sqrt((sinxcos alpha +cos x sin alpha)/(sinx))+C`
` =-(2)/(sinalpha)sqrt((sin(x+alpha))/(sinx))+C`