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The integral int(0)^(pi)sqrt(1+4"sin"^(2...

The integral `int_(0)^(pi)sqrt(1+4"sin"^(2)x/2-4"sin"x/2)dx` equals

A

`pi-4`

B

`(2pi)/3-4-sqrt(3)`

C

`4sqrt(3)-4`

D

`4sqrt(3)-4-(pi)/3`

Text Solution

Verified by Experts

The correct Answer is:
D
`I=int_(0)^(pi)sqrt(1+4"sin"^(2)x/2-4"sin"x/2)dx`
`=int_(0)^(x)|1-2"sin"x/2|dx`
`=int_(0)^(pi//3)|1-2"sin"x/2|dx`
`=int_(0)^(pi//3) (1-2"sin"x/2)dx+int_(pi//3)^(pi)(2"sin"x/2-1)dx`
`=(x+4"cos"x/2)|._(0)^(pi//3)+(-4"cos"x/2-x)|_(pi//3)^(pi)`
`=4sqrt(3)-4-(pi)/3`
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