Evaluate the integrals `int0pi/2sqrt(sinvarphi)cos^5varphi""dvarphi`

Explanation

`"Let " sinphi =t`
` rArr cos phi d phi =dt`
`at phi=0 t=sin 0=0`
` at phi =(pi)/(2) t=sin .(pi)/(2)=1`

`:. int_(0)^(pi//2) sqrt(sin phi). cos^(5) phi dphi`
`=int_(0)^(pi//2)sqrt(sin)phi.(1-sin^(2)phi)^(2).cos phidphi`
` =int_(0)^(1) sqrt(t)(1-t^(2))^(2)dt`
`=int_(0)^(1)(t^(1//2)-2t^(5//2)+t^(9//2))dt`
`=[(2)/(3)t^(3//2)-(4)/(7)t^(7//2)+(2)/(11)t^(11//2)]_(0)^(1)`
`=(2)/(3)-(4)/(7) +(2)/(11)`
`=(154-132+42)/(231)=(64)/(231)`