Integral of odd powered trig function with u substitution

TL;DR

The video explains how to find the indefinite integral of sine squared x cosine to the third x using u-substitution.

Transcript

• [Voiceover] Let's see if we can take the indefinite integral of sine squared x cosine to the third x dx. Like always, pause the video and see if you can work it through on your own. All right, so right when you look at it you're like, "Oh wow, if this was just a sine of x, "not a sine squared of x. "Well that's going to be the negative of "the de... Read More

Key Insights

• 🦕 Indefinite integrals involving odd exponents can be simplified using algebraic manipulation and the Pythagorean Identity.
• 😄 U-substitution is a useful technique for simplifying integrals and finding the antiderivative.
• 😄 The expression can be rewritten in terms of sin^2x using the Pythagorean Identity before applying u-substitution.

Questions & Answers

Q: What is the purpose of algebraically engineering the expression when dealing with odd exponents?

Algebraically engineering the expression allows us to rewrite it in a form where we can use u-substitution, which simplifies the integration process and makes it easier to find the indefinite integral.

Q: How does the use of the Pythagorean Identity simplify the expression?

By using the Pythagorean Identity (cos^2x = 1 - sin^2x), we can rewrite the expression in terms of sin^2x and simplify it before applying u-substitution.

Q: Why is u-substitution used in this case?

U-substitution is used to simplify the expression even further by replacing sin(x) with u and its derivative, cos(x) dx, with du. This allows us to find the indefinite integral using known formulas.

Q: What is the final result of finding the indefinite integral?

The final result of finding the indefinite integral is (sin^3(x))/3 - (sin^5(x))/5 + c, where c is the constant of integration.

Summary & Key Takeaways

• The video teaches how to find the indefinite integral of sine squared x cosine to the third x using u-substitution.

• It explains the technique of algebraically engineering the expression to use u-substitution when dealing with odd exponents.

• By separating out one of the cosine x terms and using the Pythagorean Identity, the integral can be rewritten and simplified before applying u-substitution.