Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(2θ) cos³(2θ) dθ

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Start by recognizing the integral: \(\int_0^{\frac{\pi}{2}} \sin^2(2\theta) \cos^3(2\theta) \, d\theta\). Notice that the powers of sine and cosine are both positive integers, which suggests using trigonometric identities or substitution.

Use the substitution \(u = 2\theta\), which implies \(du = 2 \, d\theta\) or \(d\theta = \frac{du}{2}\). Also, change the limits of integration accordingly: when \(\theta = 0\), \(u = 0\); when \(\theta = \frac{\pi}{2}\), \(u = \pi\).

Rewrite the integral in terms of \(u\): \(\int_0^{\pi} \sin^2(u) \cos^3(u) \cdot \frac{1}{2} \, du = \frac{1}{2} \int_0^{\pi} \sin^2(u) \cos^3(u) \, du\).

Express \(\cos^3(u)\) as \(\cos(u) \cdot \cos^2(u)\) and use the Pythagorean identity \(\cos^2(u) = 1 - \sin^2(u)\) to rewrite the integral in terms of \(\sin(u)\): \(\frac{1}{2} \int_0^{\pi} \sin^2(u) \cos(u) (1 - \sin^2(u)) \, du\).

Use the substitution \(t = \sin(u)\), so \(dt = \cos(u) \, du\). The integral becomes \(\frac{1}{2} \int_{t=\sin(0)}^{t=\sin(\pi)} t^2 (1 - t^2) \, dt\). Then, expand the integrand and prepare to integrate term-by-term.

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