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int(0)^(pi)|cosx|^(3) dx is equal to ...

`int_(0)^(pi)|cosx|^(3) dx` is equal to (a) `(4)/(3)` (b) `(2)/(3)` (c) `0` (d) `(-8)/(3)`

A

`(2)/(3)`

B

`-(4)/(3)`

C

0

D

`(4)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
D
We know , graph of y = cos x is

`:.` The graph of y= | cos x | is

`I=int_(0)^(pi)|cosx|^(3)=2int_(0)^(pi/2)|cosx|^(3) dx`
(`:' y =|cosx | " is symmetric about " x=(pi)/(2))`
`=2int_(0)^(pi/(2))cos^(3)x dx [:'cosx ge0 " for " x in[0,(pi)/(2)]]`
Now , as cos `3x=4cos^(3)x-3cosx`
`:.cos^(3)x=(1)/(4)(cos3x+3cosx)`
`:.I=(2)/(4)int_(0)^(pi/(2))(cos3x+3cosx)dx`
`=(1)/(2)[(sin3x)/(3)+3 sinx]_(0)^(pi/(2))`
`=(1)/(2){[(1)/(2)sin(3pi)/(2)+3sin(pi)/(2)]-[(1)/(3)sin0+3sin0]}`
`=(1)/(2){[(1)/(3)(-1)+3]-[0+0]}`
[`:'"sin"(3pi)/(2)sin(pi+(pi)/(2))=-"sin"(pi)/(2)=-1]`
`=(1)/(2)[-(1)/(2)+3]=(4)/(3 )`
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