Integral of sin(2x)cos(2x)

TL;DR

This content explains how to integrate the product of sine and cosine using a substitution method.

Transcript

in this example we're going to integrate sine of 2x times the cosine of 2x let's go ahead and work through this solution so there's actually various ways to do this problem you can use a trig identity right from the beginning um you can make a u substitution uh and there's various u substitutions you can make let's try to think here if we let u be ... Read More

Key Insights

• 💨 There are multiple ways to integrate sine(2x) * cosine(2x), including using trigonometric identities and substitutions.
• 💦 Letting u be equal to sine(2x) simplifies the integration process significantly.
• 🎅 The final solution is (1/4) * sin^2(2x) + C, where C represents the constant of integration.

Questions & Answers

Q: What are the different approaches to integrating sine(2x) * cosine(2x)?

The video mentions that possible approaches include using a trigonometric identity, making a u-substitution, or choosing u as sine or cosine. However, letting u be equal to sine(2x) is the most straightforward approach.

Q: How does the derivative of sine(2x) relate to the substitution method?

The derivative of sine(2x) involves the cosine function, making it a suitable substitution. This choice simplifies the integration process.

Q: What does the substitution transform sine(2x) * cosine(2x) into?

After making the substitution, the integration becomes (1/2) * du, where u represents sine(2x).

Q: How do we find the final solution after making the substitution?

By integrating (1/2) * du, we apply the power rule to obtain (1/4) * sin^2(2x) + C, where C is the constant of integration.

Summary & Key Takeaways

• The video discusses different approaches to integrate sine(2x) * cosine(2x).

• The best approach is to let u be equal to sine(2x) and use the derivative chain rule.

• After making the substitution, the integration simplifies to (1/4) * sin^2(2x) + C.