`(1)/(sqrt(sin^(3)xsin(x+a)))`

Explanation

माना `I=int(1)/(sqrt(sin^(3)xsin(x+alpha)))dx`
`I=int(1)/(sqrt(sin^(3)x(sinxcos alpha+cosxsinalpha)))dx`
`=int(1)/(sqrt(sin^(4)x(cos alpha+cotx sinalpha)))dx`
`=int(1)/(sin^(4)x(cosalpha+cot x sinalpha))dx`
`=int(1)/(sin^(2)xsqrt(cos alpha+cotx sin alpha))dx`
`=int("cosec"^(2)x)/(sqrt(cos alpha+cotx sin alpha))dx`
माना `cosalpha+cot x sin alpha=t`
`rArr" "-"cosec"^(2)x sin alphadx=dt`
`rArr" cosec"^(2)xdx=-(dt)/(sin alpha)`
`therefore I=(-1)/(sinalpha) int(dt)/(sqrtt)=(-1)/(sin alpha)(2sqrtt)+C`
`=(-1)/(sin alpha)2sqrt(cos alpha+cot x sin alpha)+C`
`=(-1)/(sin alpha)sqrt(cos alpha+(cosx sin alpha)/(sinx))+C`
`=(-2)/(sin alpha)sqrt((sinx cos alpha+cos x sin alpha)/(sin x))+C`
`=-(2)/(sin alpha)sqrt((sin(x+a))/(sinx))+C`