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The value of I=int(0)^(pi)x(sin^(2)(sinx...

The value of `I=int_(0)^(pi)x(sin^(2)(sinx)+cos^(2)(cosx))dx` is

A

`pi^(2)`

B

`(pi^(2))/(2)`

C

`(pi^(2))/(4)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
`I=int_(0)^(pi)x(sin^(2)(sinx)+cos^(2)(cosx))dx" (1)"`
`therefore" "I=int_(0)^(pi)(pi-x)(sin^(2)(sinx)+cos^(2)(cosx))dx" (2)"`
Adding (1) and (2)
`rArr" "2I=2piint_(0)^((pi)/(2))(sin^(2)(sinx)+cos^(2)(cosx))dx`
`rArr" "I=piint_(0)^((pi)/(2))(sin^(2)(sinx)+cos^(2)(cosx))dx" (3)"`
`rArr" "I=piint_(0)^((pi)/(2))(Sin^(2)(cosx)+cos^(2)(sinx))dx" (4)"`
Adding (3) and (4)
`rArr" "2I=piint_(0)^((pi)/(2))2dxrArrI=(pi^(2))/(2)`
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