9–61. Trigonometric integrals Evaluate the following integrals.
20. ∫ sin⁻³ᐟ²x cos³x dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves powers of sine and cosine. Use trigonometric identities to simplify the expression. Specifically, note that $sin^\{-3/2\}(x)$ and $cos^3(x)$ are present.
2. Step 2: Choose a substitution method. Since the powers of sine and cosine are involved, consider substituting $u\; =\; sin(x)$, which implies $du\; =\; cos(x)\; dx$. Rewrite the integral in terms of $u$.
3. Step 3: Substitute $u$ into the integral. Replace $sin^\{-3/2\}(x)$ with $u^\{-3/2\}$ and $cos(x)\; dx$ with $du$. The integral becomes $\int \; u^\{-3/2\}\; u^3\; du$.
4. Step 4: Simplify the integral. Combine the powers of $u$ to get $\int \; u^\{3\; -\; 3/2\}\; du$, which simplifies to $\int \; u^\{3/2\}\; du$.
5. Step 5: Integrate $u^\{3/2\}$ using the power rule for integration. The power rule states that $\int \; u^n\; du\; =\; (u^\{n+1\})\; /\; (n+1)\; +\; C$, where $n\; \ne \; -1$. Apply this rule to find the antiderivative, then substitute back $u\; =\; sin(x)$ 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. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas, which can help rewrite integrals in a more manageable form.

Integration Techniques

Integration techniques are methods used to find the integral of a function. For trigonometric integrals, techniques such as substitution, integration by parts, and trigonometric identities are often employed. Recognizing the appropriate technique is crucial for solving integrals efficiently, especially when dealing with products or powers of trigonometric functions.

Substitution Method

The substitution method is a technique used in integration to simplify the integral by changing variables. This involves substituting a part of the integrand with a new variable, which can make the integral easier to evaluate. In the case of trigonometric integrals, substituting a trigonometric function can often lead to a more straightforward integral that can be solved using basic integration rules.