Evaluate: (i) `int(cos^3x)/(sqrt(sinx))\ dx` (ii) `int(sin^3x)/(sqrt(cosx)\ )\ dx`

Explanation

`(i)`given that,
`intcos^3x/sqrt (sinx)dx`
`I=intcos^3x/sqrt (sinx)dx`
`I=int(cos^2x/sqrt (sinx))cosxdx`
`I=int((1-sin^2x)/sqrt (sinx))cosxdx`
Let `sinx=t`
`cosxdx=dt`
`I=int((1-t^2)/sqrt (t))dt`
`I=∫t^-(1/2)dt−∫t^(3/2)dt`
`I=(t^(1/2))/(1/2)−(t^(5/2))/(5/2)dt`
`I=2(t^(1/2)−(t^(5/2))/5)dt`
`I=2(sqrtsinx−(sqrtsin^5x)/5)+c`


`(ii)`Given that,
`int(sin^3x)/(sqrt(cosx)\ )\ dx`
`I=intsin^3x/sqrt (cosx)dx`
`I=int(sin^2x/sqrt (cosx))sinxdx`
`I=int((1-cos^2x)/sqrt (cos))sinxdx`
Let `cosx=t`
`-sinxdx=dt`
`I=-int((1-t^2)/sqrt (t))dt`
`I=-∫t^-(1/2)dt+∫t^(3/2)dt`
`I=-(t^(1/2))/(1/2)+(t^(5/2))/(5/2)dt`
`I=-2(t^(1/2)+(t^(5/2))/5)dt`
`I=2(sqrtcos−(sqrtcos^5x)/5)+c`