2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
29. ∫ cos⁴ x/sin⁶ x dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves powers of trigonometric functions. Rewrite the integrand using trigonometric identities to simplify the expression. For example, use the identity $sin²x\; +\; cos²x\; =\; 1$ if applicable.
2. Step 2: Break the integrand into separate terms or rewrite it in terms of a single trigonometric function. For instance, express $cos⁴x$ as $(cos²x)²$ and $sin⁶x$ as $(sin²x)³$.
3. Step 3: Use substitution to simplify the integral. Let $u\; =\; sinx$, which implies $du\; =\; cosx\; dx$. Rewrite the integral in terms of $u$.
4. Step 4: After substitution, simplify the resulting integral. You may need to expand or factor terms to make the integration process easier. For example, the integral might involve powers of $u$ that can be integrated directly.
5. Step 5: Once the integral is evaluated in terms of $u$, substitute back $u\; =\; sinx$ to express the solution in terms of $x$. Simplify the final expression if necessary.

Key Concepts

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric identities. Understanding these methods is crucial for simplifying complex integrals, such as those involving powers of trigonometric functions.

Trigonometric Identities

Trigonometric identities are equations that relate the angles and sides of triangles through sine, cosine, and other trigonometric functions. These identities, such as the Pythagorean identity and double angle formulas, can be used to simplify integrals involving trigonometric functions, making them easier to evaluate.

Power Reduction Formulas

Power reduction formulas are specific trigonometric identities that express powers of sine and cosine in terms of first-degree functions. For example, cos²(x) can be rewritten using the identity cos²(x) = (1 + cos(2x))/2. These formulas are particularly useful for integrals involving even powers of sine and cosine, as they help reduce the complexity of the integral.