Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ cos²(2θ) sin(θ) dθ

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1. Recognize that the integral involves \( \cos^2(2\theta) \) and \( \sin(\theta) \), and that simplifying \( \cos^2(2\theta) \) using a trigonometric identity will make the integral easier to evaluate.
2. Use the power-reduction identity for cosine squared: \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \). Applying this to \( \cos^2(2\theta) \) gives \( \cos^2(2\theta) = \frac{1 + \cos(4\theta)}{2} \).
3. Rewrite the integral by substituting the identity: \[ \int \cos^2(2\theta) \sin(\theta) \, d\theta = \int \frac{1 + \cos(4\theta)}{2} \sin(\theta) \, d\theta. \]
4. Distribute \( \sin(\theta) \) inside the integral: \[ \int \frac{1}{2} \sin(\theta) \, d\theta + \int \frac{\cos(4\theta)}{2} \sin(\theta) \, d\theta. \]
5. Evaluate each integral separately. The first integral is straightforward, while the second integral \( \int \cos(4\theta) \sin(\theta) \, d\theta \) may require using product-to-sum identities or substitution to simplify before integrating.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They simplify expressions and integrals by transforming products or powers of sine and cosine into sums or simpler forms, such as using the power-reduction or double-angle formulas.

Integration of Trigonometric Functions

Integrating trigonometric functions involves applying standard integral formulas and substitution techniques. Recognizing when to use identities to rewrite the integrand into a more manageable form is essential for evaluating integrals involving powers or products of sine and cosine.

Substitution Method in Integration

The substitution method simplifies integrals by changing variables to reduce complexity. When the integrand contains composite functions like cos²(2θ), substituting an inner function or using identities can transform the integral into a basic form that is easier to evaluate.