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int(0)^(pi//2)(sqrt(sinx)cosx)^(3)dx=?...

`int_(0)^(pi//2)(sqrt(sinx)cosx)^(3)dx=?`

A

`(2)/(9)`

B

`(2)/(15)`

C

`(8)/(45)`

D

`(5)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
C
`I=int_(0)^(pi//2)(sinx)^(3//2)cos^(2)xcosxdx=int_(0)^(1)t^(3//2)(1-t^(2))dt`, where `sinx=t`
`int_(0)^(1)(t^(3//2)-t^(7//2))dt=[(2)/(5)t^(5//2)-(2)/(9)t^(9//2)]_(0)^(1)=((2)/(5)-(2)/(9))=(8)/(45)`.
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