`int_(0)^(pi//2)sqrt(costheta)sin^(3)thetad theta` का मान है

Options

A. `(8)/(21)`
B. `-(8)/(21)`
C. `-(20)/(21)`
D. इनमें से कोई नहीं

Answer

A

Explanation

`I=int_(0)^(pi//2)sqrt(costheta)(1-cos^(2)theta)sin thetad theta`
माना `costheta=t`, तो `-sinthetad theta=dt`
`thereforeI=-int_(1)^(0)sqrt(t)(1-t^(2))dt`
`=int_(0)^(1)(sqrt(t)-t^(5//2))dt`
`=[(2)/(3)t^(3//2)-(2)/(7)t^(7//2)]_(0)^(1)=(2)/(3)-(2)/(7)=(8)/(21)`