Integral of sin^5(x)

TL;DR

Learn how to integrate sine to the fifth power using substitution and splitting it into two parts.

Transcript

in this video we're going to talk about how to integrate sine to the fifth power of X DX so the first thing that we need to do is we need to split sine to the fifth power into two different parts and that is sine to the fourth times sine and is a invisible one here so 4 plus 1 is 5 now the reason why we need to do this is because you need to use us... Read More

Key Insights

• ✊ Odd powers in integrals require us to keep one of the trig functions on the outside.
• 💦 Substitution, like letting u equal cosine X, can simplify integrals involving trigonometric functions.
• 😑 Expanding and simplifying the expression before finding the antiderivative is a crucial step in integration.

Questions & Answers

Q: Why do we need to split sine to the fifth power into two parts?

Splitting sine to the fifth power into sine to the fourth power times sine allows us to use substitution and keep one of the trig functions on the outside, which is necessary when dealing with odd powers.

Q: What is the result of replacing sine squared with 1 minus cosine squared?

By replacing sine squared with 1 minus cosine squared, we obtain 1 minus u squared squared as part of the expression during the integration.

Q: How do we perform u substitution in this case?

To perform u substitution, we let u equal cosine X, resulting in du divided by negative sine X for the substitution of dx.

Q: What is the final answer to the integral of sine raised to the fifth power?

The final answer is -1/5 cosine to the fifth power of X plus 2/3 cosine to the third power of X minus cosine X plus a constant term.

Summary & Key Takeaways

• Split sine to the fifth power into sine to the fourth power times sine.

• Use u substitution by letting u equal cosine X to simplify the integral.

• Expand the expression, simplify, and find the antiderivative of each term.

• Replace u with cosine X and write the expression in standard form to obtain the final answer.