To solve the integral of sin(2x) cos(2x), we can use the identity:
sin(2x) = 2 sin(x) cos(x). This means:
sin(2x) cos(2x) = (1/2) sin(4x).
Thus, we can rewrite the integral:
∫ sin(2x) cos(2x) dx = (1/2) ∫ sin(4x) dx.
The integral of sin(4x) is:
= -1/4 cos(4x) + C.
Now, substituting this back into the integral:
∫ sin(2x) cos(2x) dx = -1/8 cos(4x) + C.
We need to evaluate this from x = 1 to x = 2, and set the result equal to 1/2 sin(2x).
Calculating:
F(2) = -1/8 cos(4 * 2) = -1/8 cos(8)
F(1) = -1/8 cos(4 * 1) = -1/8 cos(4)
Then, the definite integral is:
F(2) - F(1) = -1/8 cos(8) + 1/8 cos(4) = (1/2) (sin(4) - sin(2)).
Thus, we need to solve for a and b which correspond to these values. After evaluating numeric values, we find:
a = 1, b = 2.
So the final answer is:
a = 1, b = 2.