Integral of sin(3x)*sin(2x) using the Product to Sum Formula Trigonometric Integrals Calculus 2 | Summary and Q&A

TL;DR

Learn how to integrate the product of two sines using a trigonometric identity and simplify the result.

Key Insights

• 👨‍💼 The product of two sines can be integrated using a trigonometric identity involving cosine.
• 🥳 Breaking down the integral into separate parts simplifies the integration process.
• 🥳 The derivative of sine is cosine, which is used to integrate the separate parts of the integral.
• 🗂️ Coefficients in front of the cosine can be divided by when integrating.
• ❓ Understanding the formulas and identities is crucial in solving integration problems.
• ❓ Integrating trigonometric functions often involves recognizing patterns and applying relevant identities.
• 💦 Working with trigonometric identities can make integration faster and more efficient.

Transcript

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Questions & Answers

Q: How do you integrate the product of two sines?

Use the trigonometric identity: sine of A times sine of B equals 1/2 times cosine of (A-B) minus cosine of (A+B).

Q: What are the values of A and B in the given problem?

In the given problem, A is 3x and B is 2x.

Q: How can the integral be simplified after applying the trigonometric identity?

The integral can be simplified by breaking it into separate integrals and applying the derivative of cosine to integrate each part.

Q: How do you integrate cosine with a coefficient?

If you have an integral of the form cosine of (Bx) with B not equal to 0, integrate it to get sine of (Bx) and divide by the coefficient B.

Summary & Key Takeaways

• To integrate the product of two sines, use the trigonometric identity: sine of A times sine of B equals 1/2 times cosine of (A-B) minus cosine of (A+B).

• Apply the identity to the given problem, which is integrating sine 3x times sine 2x.

• Break down the integral into separate parts and integrate each using the derivative of cosine and the specific formula for integrating cosine with a coefficient.