The integral `int_0^pisqrt(1+4sin^2(x/2)-4sin(x/2)dx)`equal(1) `pi-4`(2) `(2pi)/3-4-4sqrt(3)`(3) `4""sqrt(3)-4`(4) `4""sqrt(3)-4-pi/3`

The integral `int_0^pisqrt(1+4sin^2(x/2)-4sin(x/2)dx)`equal(1) `pi-4`(

1.	The integral `int_0^pisqrt(1+4sin^2(x/2)-4sin(x/2)dx)`equal(1) `pi-4`(2) `(2pi)/3-4-4sqrt(3)`(3) `4""sqrt(3)-4`(4) `4""sqrt(3)-4-pi/3`
Answer» `i_1 = int_0^pi sqrt(1 + 4sin^2 x/2 - 4sin x/2 )dx` `= int_0^pi sqrt((1)^2 + (2sin (x/2))^2 - 2 xx1xx 2sin(x/2))dx` `= int_0^pi sqrt((1-2sin(x/2))^2) dx` `= int_0^pi \| 1- 2sin (x/2)\|dx` `sin (x/2)= 1/2` `x/2 = pi/6` `x= pi/3` `int_0^(pi/3) (1- 2sin(x/2) dx + int_(pi/3)^pi ( 2 sin(x/2)-1) dx` `= (x+4cos(x/2))\| + (-4cos(x/2) - x)\|_(pi/3)^pi` `=[(pi/3 + 4cos (pi/6) ) - (0+4coso) + [(-4cos(pi/2) - pi)- (-4cos(pi/6)]` `= [pi/3 + 4 xx sqrt3/2 - 4] + [ - pi + 4 sqrt 3/2+ pi/3]` `= pi/3 + 4 xx sqrt3/2 - 4 - pi+ 4 sqrt3/2 + pi/3` `= 4 sqrt3 - 4 - pi/3` Answer