Question Detail

Let's use u-substitution for the integral. Let u = cos(2x).Then, du/dx = -sin(2x) * 2 = -2 sin(2x). So, sin(2x) dx = -1/2 du.The integral becomes ∫ 4 * u⁴ * (-1/2) du = -2 ∫ u⁴ du.Integrating gives: -2 * (u⁵/5) + C = -(2/5) cos⁵(2x) + C.Now, evaluate the definite integral from 0 to θ:[-(2/5) cos⁵(2x)]₀^θ = -(2/5) cos⁵(2θ) - (-(2/5) cos⁵(2*0))= -(2/5) cos⁵(2θ) + (2/5) cos⁵(0)= -(2/5) cos⁵(2θ) + (2/5) * 1⁵ = (2/5) (1 - cos⁵(2θ)).We are given that this integral equals 4/5:(2/5) (1 - cos⁵(2θ)) = 4/5Divide both sides by 2/5: 1 - cos⁵(2θ) = (4/5) * (5/2) = 2.cos⁵(2θ) = 1 - 2 = -1.Taking the fifth root of both sides: cos(2θ) = -1.For the given range 0 < θ < π, the range for 2θ is 0 < 2θ < 2π.The value of φ in (0, 2π) for which cos(φ) = -1 is φ = π.So, 2θ = π => θ = π/2.