9–61. Trigonometric integrals Evaluate the following integrals.
22. ∫[π/4 to π/2] sin²(2x) cos³(2x) dx

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 identity $sin²(x)\; =\; 1\; -\; cos²(x)$ or consider substitution methods for products of sine and cosine.
2. Step 2: Let $u\; =\; cos(2x)$. Then, compute the derivative of $u$ with respect to $x$, which gives $du\; =\; -2sin(2x)dx$. Rewrite the integral in terms of $u$.
3. Step 3: Adjust the limits of integration. When $x\; =\; \pi /4$, $u\; =\; cos(\pi /2)\; =\; 0$. When $x\; =\; \pi /2$, $u\; =\; cos(\pi )\; =\; -1$. Update the integral limits accordingly.
4. Step 4: Substitute $sin²(2x)$ and $cos³(2x)$ in terms of $u$. The integral becomes a polynomial in $u$, which is easier to evaluate.
5. Step 5: Integrate the resulting polynomial with respect to $u$, and then evaluate the definite integral using the updated limits of integration.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. They are essential for simplifying integrals involving trigonometric functions. For example, the identity sin²(x) + cos²(x) = 1 can be used to rewrite integrals in a more manageable form.

Integration Techniques

Integration techniques, such as substitution and integration by parts, are methods used to evaluate integrals that may not be straightforward. In the case of the integral ∫ sin²(2x) cos³(2x) dx, substitution can simplify the expression by letting u = sin(2x) or cos(2x), making the integral easier to solve.

Definite Integrals

Definite integrals calculate the area under a curve between two specified limits, in this case, from π/4 to π/2. The result of a definite integral is a numerical value that represents this area. Understanding how to evaluate definite integrals is crucial for finding the total accumulation of a quantity represented by the integrand over the given interval.