The integral is equal to: #03

• Integration

The integral ₀∫^π √(1 + 4 sin²x/2 - 4 sinx/2) dx equals

1. 4√3 - 4
2. 4√3 - 4 - π/3
3. 2π/3 - 4 - 4√3
4. π - 4

Answer

₀∫^π (√1 + 4 sin2x/2 - 4 sinx/2) dx

= ₀∫^π |1 - 2sin(x/2)| dx

= ₀∫^π/3 (1 - 2sin(x/2)) dx + _π/3∫^π (2sin(x/2) - 1) dx

= (x + 4 cos(x/2)₀ ^π/3) + (-4 cos(x/2) - x)_π/3^π

The correct option is B.