9–61. Trigonometric integrals Evaluate the following integrals.
13. ∫ sin⁵x dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves an odd power of sine, which suggests using the identity $sin²x\; =\; 1\; -\; cos²x$ to simplify the expression. Split $sin⁵x$ as $sin⁴x\; ·\; sinx$.
2. Step 2: Rewrite $sin⁴x$ using the identity $sin²x\; =\; 1\; -\; cos²x$. This gives $sin⁴x\; =\; (sin²x)²\; =\; (1\; -\; cos²x)²$.
3. Step 3: Substitute $sin⁴x$ into the integral, resulting in $\int \; (1\; -\; cos²x)²\; ·\; sinx\; dx$. This prepares the integral for a substitution.
4. Step 4: Use the substitution $u\; =\; cosx$, which implies $du\; =\; -sinx\; dx$. Replace $sinx\; dx$ with $-du$ and rewrite the integral in terms of $u$.
5. Step 5: Expand $(1\; -\; u²)²$ to get $1\; -\; 2u²\; +\; u⁴$, and integrate term by term. After integration, substitute back $u\; =\; cosx$ to express the result in terms of $x$.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus, particularly in integration. The sine function, sin(x), oscillates between -1 and 1 and is periodic with a period of 2π. Understanding the properties of these functions is essential for evaluating integrals involving them, as they often require specific techniques or identities.

Integration Techniques

Integration techniques are methods used to find the integral of a function. For trigonometric integrals, techniques such as substitution, integration by parts, or using trigonometric identities are commonly employed. In the case of ∫ sin⁵x dx, recognizing patterns and applying the appropriate technique is crucial for simplifying the integral.

Power Reduction Formulas

Power reduction formulas are used to express higher powers of trigonometric functions in terms of lower powers. For example, the formula for sin²x can be expressed as (1 - cos(2x))/2. This is particularly useful for integrals like ∫ sin⁵x dx, as it allows the integral to be rewritten in a more manageable form, facilitating easier evaluation.