Evaluate the definite Integral
Integral [0 to pi/4] cos(2x)sec^2(pi/4 sin(2x))dx
Let u = tan(π/4 sin(2x))
du = sec^2(π/4 sin(2x))*(π/4)(2cos(2x)) du
So, what you have is just ∫du with some constant factored out.
To evaluate the definite integral ∫[0 to π/4] cos(2x)sec^2(π/4 sin(2x))dx, you can follow these steps:
Step 1: Simplify the integrand functions.
Use trigonometric identities to simplify the expression. Start by replacing sec^2(θ) with 1 + tan^2(θ) and simplify from there.
sec^2(π/4 sin(2x)) = 1 + tan^2(π/4 sin(2x))
Step 2: Simplify further.
Using the double-angle identity for sine, sin(2θ) = 2sin(θ)cos(θ), we can rewrite the expression as:
π/4 sin(2x) = π/4 * 2sin(x)cos(x) = π/2 sin(x)cos(x)
so:
tan^2(π/4 sin(2x)) = tan^2(π/2 sin(x)cos(x))
Step 3: Substitute the simplified expressions into the integral.
Now we can substitute the simplified expressions into the original integral:
∫[0 to π/4] cos(2x)(1 + tan^2(π/4 sin(2x))) dx
= ∫[0 to π/4] cos(2x)(1 + tan^2(π/2 sin(x)cos(x))) dx
Step 4: Use a trigonometric substitution.
Let u = π/2 sin(x)cos(x). Then, du = (π/2)[cos^2(x) - sin^2(x)] dx.
We can rewrite the integral using the new variable u:
= (2/π) ∫[0 to π/4] (1 + tan^2(u)) du
= (2/π) ∫[0 to π/4] (sec^2(u)) du
Step 5: Evaluate the integral.
Now, we can evaluate the integral ∫(sec^2(u)) du.
∫(sec^2(u)) du = tan(u) + C, where C is the constant of integration.
Therefore, the original definite integral becomes:
(2/π) [tan(u)] from 0 to π/4
Step 6: Evaluate the definite integral by plugging in the limits.
Now, substitute the limits into the expression:
= (2/π) [tan(π/4) - tan(0)]
= (2/π) [(1) - (0)]
= 2/π
Therefore, the value of the definite integral ∫[0 to π/4] cos(2x)sec^2(π/4 sin(2x))dx is 2/π.