9–61. Trigonometric integrals Evaluate the following integrals.
26. ∫ sin³x cos³ᐟ²x dx

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Rewrite the integral using trigonometric identities. Specifically, use the identity for powers of sine and cosine: \( \sin^3(x) = \sin(x) \cdot \sin^2(x) \) and \( \sin^2(x) = 1 - \cos^2(x) \). This transforms the integral into \( \int \sin(x) (1 - \cos^2(x)) \cos^{3/2}(x) \, dx \).

Perform a substitution to simplify the integral. Let \( u = \cos(x) \), which implies \( du = -\sin(x) \, dx \). Substitute these into the integral, replacing \( \sin(x) \, dx \) with \( -du \) and \( \cos(x) \) with \( u \). The integral becomes \( -\int (1 - u^2) u^{3/2} \, du \).

Expand the integrand \( (1 - u^2) u^{3/2} \) by distributing \( u^{3/2} \). This gives \( -\int (u^{3/2} - u^{7/2}) \, du \).

Integrate each term separately. Use the power rule for integration, \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), to find the antiderivative of \( u^{3/2} \) and \( u^{7/2} \).

After finding the antiderivative, substitute back \( u = \cos(x) \) to return to the original variable. Simplify the expression to complete the solution.

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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. Understanding these identities helps in transforming complex integrals into simpler forms that are easier to evaluate.

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 the use of trigonometric identities are often employed. In the case of products of sine and cosine functions, using identities to express the integrand in a more manageable form can significantly simplify the integration process.

Power Reduction Formulas

Power reduction formulas are specific trigonometric identities that allow us to express powers of sine and cosine in terms of first-degree functions. For example, sin²x can be rewritten using the identity sin²x = (1 - cos(2x))/2. These formulas are particularly useful when integrating higher powers of sine and cosine, as they reduce the degree of the functions, making the integral easier to solve.

9–61. Trigonometric integrals Evaluate the following integrals.26. ∫ sin³x cos³ᐟ²x dx