9–61. Trigonometric integrals Evaluate the following integrals.
25. ∫ sin²x cos⁴x dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves powers of sine and cosine. For integrals of this type, it is often helpful to use trigonometric identities to simplify the expression. Specifically, use the Pythagorean identity: $sin²x\; =\; 1\; -\; cos²x$ or $cos²x\; =\; 1\; -\; sin²x$, depending on the situation.
2. Step 2: Split the powers of sine and cosine into manageable parts. For example, rewrite $sin²x\; cos⁴x$ as $sin²x\; (cos²x)²$. This will allow us to work with smaller powers and apply substitution techniques.
3. Step 3: Use substitution to simplify the integral. Let $u\; =\; cosx$, which implies $du\; =\; -sinx\; dx$. Rewrite the integral in terms of $u$ and $du$. The integral becomes $\int \; sin²x\; cos⁴x\; dx\; =\; \int \; (1\; -\; u²)\; u⁴\; (-du)$.
4. Step 4: Simplify the integral further. Distribute the terms and simplify: $\int \; (1\; -\; u²)\; u⁴\; (-du)\; =\; -\int \; (u⁴\; -\; u⁶)\; du$. Now, integrate each term separately using the power rule for integration: $\int \; uⁿ\; du\; =\; uⁿ⁺¹\; /\; (n\; +\; 1)$.
5. Step 5: After integrating, substitute back $u\; =\; cosx$ to return the solution to the original variable $x$. Combine the terms and include the constant of integration $C$.

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 sin²x and cos⁴x, 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, sin²x can be rewritten using the identity sin²x = (1 - cos(2x))/2. These formulas are particularly useful in integrals, as they transform the integrand into a more manageable form, facilitating easier integration.