Prove that : `int_0^(pi/2)sqrt(1-sin2x)dx=2(sqrt(2)-1)`

Options

A. `2sqrt(2)`
B. `2(2+sqrt(2))`
C. `2`
D. `2(sqrt(2)-1)`

Answer

A

Explanation

Let `I = int_(0)^(pi//2) sqrt(1-sin2x)d`
` int_(0)^(pi//4) sqrt((cosx - sinx)^(2)) dx + int_(pi//4)^(pi//2) sqrt((sinx-cosx)^(2))dx`
`=[sinxcosx]_(0)^(pi//4)+[-cosx-sinx]_(pi//4)^(pi//2)`
`= 1/(sqrt(2)) + 1/(sqrt(2)) - 0 - 1 + (-0-1+1/(sqrt(2)) + (1)/(sqrt(2)))`
`= 2sqrt(2)-2=2(sqrt(2)-1)`