if `I=int_(0)^(pi//2)(3sqrt(costheta))/((sqrt(sintheta)+sqrt(costheta))^(5))d theta`, then `I^(2)` is equal to

Answer

0.25

Explanation

`I=3underset(0)overset(pi//2)int(sqrt(costheta)d theta)/((sqrt(sintheta)+sqrt(costheta))^(5))=3underset(0)overset(pi//2)int(3sqrt(sintheta^(2))d theta)/((sqrt(costheta)+sqrt(sintheta))^(5))`
`impliesunderset(0)overset(pi//2)int(3sqrt(costheta)d theta)/((sqrt(sintheta)+sqrt(costheta))^(5))=3underset(0)overset(pi//2)int(sqrt(sintheta) d theta)/((sqrt(costheta)+sqrt(sintheta))^(5))implies2l=3underset(0)overset(pi//2)int(d theta)/((sqrt(sintheta)+sqrt(costheta))^(4))`
`(2I)/(3)=underset(0)overset(pi//2)int(sec^(2)theta d theta)/((sqrt(tantheta)+1)^(4))`
let `tantheta=t^(2)impliesec^(2)theta d theta=2tdt`
`implies(2I)/(3)=underset(0)overset(infty)int(2tdt)/((t+1)^(4))implies(I)/(3)=underset(0)overset(infty)int[(1)/((t+1)^(3))-(1)/((t+1)^(4))]dt`
`impliesI=|(-3)/(2(t+1)^(2))+(1)/((t+1)^(3))|_(0)^(infty)=(3)/(2)-1=(1)/(2)`