Question

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.) \(\int \sin ^{3} x \cos ^{3} x d x\)

Short answer

Use trigonometric identities and substitution to integrate, resulting in \(\frac{1}{32}\sin^4 2x + C\).

Step-by-step solution

Simplify the Integrand

To solve the integral \(\int \sin^3 x \cos^3 x \; dx\), first observe the powers of sine and cosine. Both are raised to odd powers, so we can use trigonometric identities to simplify. Set aside one sine and cosine term, leaving an even power for the other terms: \(\sin^3 x \cos^3 x = \sin x \cos x \cdot \sin^2 x \cos^2 x\).

Use Trigonometric Identity

Use the identity \(\sin^2 x = 1 - \cos^2 x\) (or the equivalent identity for cosine) to simplify further: \(\sin^2 x \cos^2 x = (1 - \cos^2 x)(1 - \sin^2 x)\). This allows us to express the integrand in terms of a single trigonometric function, either sine or cosine.

Substitute and Convert

Express \(\sin x \cos x\) using the identity \(\sin x \cos x = \frac{1}{2} \sin 2x\). Substitute this in the integral: \(\int (\frac{1}{2} \sin 2x)(1-\cos^2 x)(1-\sin^2 x)\; dx\).

Distribute and Simplify

Expand the expression \((1 - \cos^2 x)(1 - \sin^2 x)\) to \(1 - \cos^2 x - \sin^2 x + \cos^2 x \sin^2 x\). Integral becomes \(\frac{1}{2} \int \sin 2x (1 - \cos^2 x - \sin^2 x + \cos^2 x \sin^2 x)\; dx\).

Integrate Term-by-Term

Compute each term separately. For instance, \(\int \sin 2x\; dx = -\frac{1}{2} \cos 2x\), and for others apply substitutions, if necessary. For example, \(u = \cos x\), then \(du = -\sin x dx\) for appropriate terms.

Verify with CAS

Finally, use a Computer Algebra System (CAS) to verify the computed integral matches: \(\int \sin^3 x \cos^3 x \; dx = \frac{1}{32} \sin^4 2x\).

Key concepts

Integration Techniques

When dealing with integrals of trigonometric functions, it's important to understand which integration techniques to apply. In this case, the integral involves powers of sine and cosine, spreading across the equation \(\int \sin^3 x \cos^3 x \; dx\). Here are some techniques used:

• Splitting: We start by breaking down the powers into products of smaller functions. By setting aside one factor each of \(\sin x\) and \(\cos x\), we simplify the remaining even powers to ensure easier integration.
• Substitution: Using identities like \(\sin^2 x = 1 - \cos^2 x\), we express the functions in terms of a single variable, reducing complexity for easier integration.
• Term-by-term Integration: Once simplified, the integral splits into separate terms which can be integrated individually, often leveraging known integral results, like \(\int \sin 2x\; dx = -\frac{1}{2} \cos 2x\).

Each technique aims to make solving the integral more manageable by reducing function complexity and focusing on simpler components of the equation.

Trigonometric Identities

Trigonometric identities are key tools in simplifying complex trigonometric expressions. In the context of the integral \(\int \sin^3 x \cos^3 x \; dx\), several identities are pivotal:

• Pythagorean Identity: Identities like \(\sin^2 x = 1 - \cos^2 x\) and \(\cos^2 x = 1 - \sin^2 x\) are indispensable for rewriting terms in a different trigonometric perspective, eventually simplifying calculations.
• Double Angle Identity: The identity \(\sin x \cos x = \frac{1}{2} \sin 2x\) helps express products of sine and cosine in terms of a double angle, providing a simpler expression for integration.

By applying these identities, the integrand's complexity is reduced, aligning it with known integral forms. This strategic rewriting transforms challenging integrals into familiar patterns, facilitating a straightforward integration process.

Computer Algebra System (CAS)

When solving complex integrals like \(\int \sin^3 x \cos^3 x \; dx\), checking solutions with a Computer Algebra System (CAS) can be invaluable. CAS tools, like Mathematica, Maple, or MATLAB, offer numerous benefits:

• Solution Verification: By comparing manually derived solutions with those generated by CAS, one can confirm correctness and address discrepancies. This serves as a reliable mechanism for ensuring accuracy.
• Integral Solving: CAS can provide step-by-step solutions for integrals, breaking down each process. This is particularly beneficial for understanding and learning new techniques one might apply manually.
• Learning Assistance: By analyzing CAS outputs, students can pick up new methods and tricks, seeing problems solved from various angles and sometimes simpler than expected.

Leveraging CAS tools can significantly enhance one's understanding, offering a complementary aid to traditional problem-solving methods and ensuring precise outcomes.