Evaluate exactly, using the Fundamental Theorem:
∫^(π/4)_01cos2(x)dx=
We begin by using the antiderivative of cos^2(x), which is (1/2)(x + sin(2x)/2), as obtained by using the double angle formula for cosine and integrating term by term:
∫ cos^2(x)dx = ∫ (1/2)(1 + cos(2x))dx = (1/2)(x + (1/2)sin(2x)) + C,
where C is the constant of integration.
Using this, we can evaluate the definite integral by plugging in the bounds and subtracting:
∫^(π/4)_01cos^2(x)dx = [(1/2)(π/4 + (1/2)sin(π/2)) - (1/2)(0 + (1/2)sin(0))]
= (1/2)(π/4 + 1)
= π/8 + 1/2.
Therefore, the exact value of the integral is π/8 + 1/2.