Question

Integrate the function:

`1/sqrt(sin^3 x sin(x + alpha))`

Answer 1

Let `I = int 1/ (sqrt(sin^3 x sin (x + alpha)))  dx`

`= int sqrt ((sinx)/(sin^4 x sin (x + alpha)))  dx`

`= int 1/ (sin^2 x) sqrt((sinx)/ (sin (x + alpha)))  dx`

Let `(sin (x + alpha))/ sinx = t`

⇒ `(sin x cos (x + alpha) - cos x sin (x + alpha))/sin^2 x  dx = dt`

⇒ `(sin [x - (x + alpha)])/sin^2 x  dx = dt`

⇒ `-(sin alpha)/sin^2 x  dx = dt`

∴ `I = int - 1/ (sin alpha)* 1/sqrtt dt`

`= -1/ (sin alpha)  int t^(-1/2)  dt`

`= -1/ (sin alpha) t^(1/2)/(1/2) + C`

`= (-2)/ (sin alpha) sqrtt + C`

`= (-2)/(sin alpha)  sqrt (sin(x + alpha)/sinx) + C`