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Question
Integrate the function:
`1/sqrt(sin^3 x sin(x + alpha))`
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Solution
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`
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