9–61. Trigonometric integrals Evaluate the following integrals.
19. ∫[0 to π/3] sin⁵x cos⁻²x dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves powers of sine and cosine. To simplify, consider using trigonometric identities or substitution methods. For example, you can use the identity $sin²x\; +\; cos²x\; =\; 1$ if needed.
2. Step 2: Rewrite the integral to make it more manageable. Since the integrand is $sin⁵x\; cos⁻²x$, you can split $sin⁵x$ into $sin³x\; sin²x$. Then, use the identity $sin²x\; =\; 1\; -\; cos²x$ to express it in terms of cosine.
3. Step 3: Perform substitution. Let $u\; =\; cosx$, which implies $du\; =\; -sinx\; dx$. Adjust the limits of integration accordingly: when $x\; =\; 0$, $u\; =\; cos(0)\; =\; 1$, and when $x\; =\; \pi /3$, $u\; =\; cos(\pi /3)\; =\; 1/2$.
4. Step 4: Substitute into the integral. Replace $sinx\; dx$ with $-du$, and express the integrand in terms of $u$. The integral becomes $\int [1\; to\; 1/2]\; (1\; -\; u²)²\; u⁻²\; (-du)$.
5. Step 5: Expand and simplify the integrand. Multiply out $(1\; -\; u²)²$ to get $1\; -\; 2u²\; +\; u⁴$. Then divide each term by $u²$ to obtain $u⁻²\; -\; 2\; +\; u²$. Integrate each term separately with respect to $u$.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus, particularly in integration. They describe relationships between angles and sides of triangles and are periodic functions. Understanding their properties, such as their ranges and symmetries, is essential for evaluating integrals involving these functions.

Integration Techniques

Integration techniques, including substitution and integration by parts, are crucial for solving complex integrals. In the case of the integral ∫ sin⁵x cos⁻²x dx, recognizing patterns and applying appropriate techniques can simplify the process. Mastery of these techniques allows for the evaluation of integrals that may not be straightforward.

Definite Integrals

Definite integrals calculate the area under a curve between two specified limits, in this case, from 0 to π/3. Understanding the properties of definite integrals, such as the Fundamental Theorem of Calculus, is vital for evaluating them. This theorem connects differentiation and integration, providing a method to compute the value of the integral using antiderivatives.