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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
PLAN Use the formual , `|x-a|={{:(x-x_(,),xgea),(-(x-a)_(,),xlta):}`
to break given integal in two parts and then integrate separately.
`int_(0)^(pi)sqrt((1-2 "sin"(x)/(2))^(2))dx=int_(0)^(pi)|1-2 "sin"(x)/(2)|dx`
`=int_(0)^(pi/(3))(1-2 "sin"(x)/(2))dx-int_(pi/(3))^(pi)(1-2 "sin" (x)/(2))dx`
`=(x+4 " cos"(x)/(4))_(0)^(pi/(3))-(x+4 " cos "(x)/(4))_(pi/(3))^(pi)`
`=4sqrt(3)-4-(pi)/(3)`
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