9–61. Trigonometric integrals Evaluate the following integrals.
16. ∫ sin²θ cos⁵θ dθ

Verified step by step guidance

1. Step 1: Recognize that the integral involves powers of sine and cosine. To simplify, use trigonometric identities. Specifically, use the Pythagorean identity: $sin²\theta \; =\; 1\; -\; cos²\theta$.
2. Step 2: Rewrite the integral by substituting $sin²\theta$ with $1\; -\; cos²\theta$. The integral becomes: $\int \; (1\; -\; cos²\theta )\; cos⁵\theta \; d\theta$.
3. Step 3: Expand the expression inside the integral. Distribute $cos⁵\theta$ to both terms, resulting in: $\int \; cos⁵\theta \; d\theta \; -\; \int \; cos⁷\theta \; d\theta$.
4. Step 4: Use the substitution method for each term. Let $u\; =\; cos\theta$, then $du\; =\; -sin\theta \; d\theta$. Rewrite the integral in terms of $u$.
5. Step 5: Solve each integral separately in terms of $u$, then substitute back $cos\theta$ to express the solution in terms of $\theta$. Combine the results to obtain the final expression.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. Key identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas. These identities can simplify integrals involving trigonometric functions, making it easier to evaluate them.

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric substitution. For integrals involving products of trigonometric functions, such as sin²θ and cos⁵θ, using substitution or recognizing patterns can significantly simplify the process.

Power Reduction Formulas

Power reduction formulas are used to express higher powers of sine and cosine in terms of first powers. For example, sin²θ can be rewritten using the identity sin²θ = (1 - cos(2θ))/2. This technique is particularly useful in integrals where the powers of trigonometric functions are high, allowing for easier integration.