9–61. Trigonometric integrals Evaluate the following integrals.
10. ∫ sin³x dx

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Step 1: Recognize that the integral involves an odd power of sine, which suggests using a trigonometric identity to simplify the expression. Rewrite sin³(x) as sin(x) * sin²(x).

Step 2: Use the Pythagorean identity sin²(x) = 1 - cos²(x) to replace sin²(x) in the integral. The expression becomes ∫ sin(x) * (1 - cos²(x)) dx.

Step 3: Perform a substitution to simplify the integral. Let u = cos(x), which implies that du = -sin(x) dx. Substitute these into the integral, transforming it into ∫ (1 - u²) (-du).

Step 4: Simplify the integral after substitution. Distribute the negative sign, resulting in ∫ (u² - 1) du.

Step 5: Integrate each term separately. The integral becomes ∫ u² du - ∫ 1 du. After integration, you will have expressions involving u. Finally, substitute back u = cos(x) to express the result in terms of x.

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Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus, representing relationships between angles and sides of triangles. The sine function, in particular, oscillates between -1 and 1 and is periodic with a period of 2π. Understanding these functions is crucial for evaluating integrals involving trigonometric expressions.

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 often employed. In the case of ∫ sin³x dx, recognizing the need to express sin³x in terms of simpler functions is essential for successful integration.

Power Reduction Formulas

Power reduction formulas are identities that help simplify the integration of powers of trigonometric functions. For sine and cosine, these formulas express higher powers in terms of first powers, making integration more manageable. For example, sin³x can be rewritten using the identity sin²x = 1 - cos²x, facilitating the evaluation of the integral.

9–61. Trigonometric integrals Evaluate the following integrals.10. ∫ sin³x dx