9–61. Trigonometric integrals Evaluate the following integrals.
15. ∫ sin³x cos²x 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, split the odd power of sine: sin³x = sin²x * sinx, and recall the Pythagorean identity sin²x = 1 - cos²x.
2. Step 2: Rewrite the integral as ∫ (1 - cos²x) * sinx * cos²x dx. This allows us to express the integral in terms of cosx, making substitution easier.
3. Step 3: Perform a substitution. Let u = cosx, which implies that du = -sinx dx. Substitute these into the integral, replacing cosx with u and sinx dx with -du.
4. Step 4: After substitution, the integral becomes ∫ (1 - u²) * u² * (-du). Simplify this expression to ∫ (-u² + u⁴) du.
5. Step 5: Integrate term by term. Use the power rule for integration: ∫ uⁿ du = uⁿ⁺¹ / (n+1). After integrating, back-substitute u = cosx to express the result in terms of x.

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 Pythagorean identity sin²x + cos²x = 1 can be used to rewrite integrals in a more manageable form.

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 the integral ∫ sin³x cos²x dx, using substitution or recognizing patterns in the integrand can simplify the process of finding the antiderivative.

Power Reduction Formulas

Power reduction formulas are used to express powers of sine and cosine in terms of first-degree functions. These formulas, such as sin²x = (1 - cos(2x))/2, help in simplifying integrals involving higher powers of trigonometric functions. Applying these formulas can make the integration of functions like sin³x cos²x more straightforward.