Evaluate: `int1/(sqrt(sin^3xsin(x+alpha)))\ dx ,\ alpha!=npi,\ \ n in Z`

Options

A. `-2"cosec" alpha(cos alpha-tanx sinalpha)^(1//2)+C`
B. `-2(cos alpha+cotx sin alpha)^(1//2)+C`
C. `-2"cosec" alpha(cos alpha-cotx sinalpha)^(1//2)+C`
D. `-2"cosec" alpha(sin alpha-cotx cos alpha)^(1//2)+C`

Answer

C

Explanation

`sin^(3)xsin(x+alpha)=sin^(3)x(sinx cos alpha + cosx sin alpha)`
`=sin^(4)x(cos alpha+cotx sin alpha)`
`I=int(1)/(sqrt(sin^(3)xsin(x+alpha)))dx`
`=int(1)/(sin^(2)xsqrt(cos alpha +cotxsin alpha))dx`
`=int("cosec"^(2)x)/(sqrt(cos alpha+cotx sin alpha))dx`
`"Putting " cos alpha+cotx sin alpha=t " and "-"cosec"^(2)xsin alphadx=dt,`
`"we have"`
`I=int-(1)/(sin alpha sqrt(t)) dt=-(1)/(sin alpha)int t^(-1//2)dt`
`=(1)/(sin alpha)((t^(1//2))/(1//2))+C`
`=-2"cosec" alpha sqrt(t)+C`
`=-2"cosec" alpha(cos alpha+cotxsin alpha)^(1//2)+C`