Integral of sin^4(x)

TL;DR

Learn how to integrate sine to the fourth power using the power reduction formula and u substitution.

Transcript

let's integrate sine to the fourth power X unfortunately we don't have the cosine X right here so we can use u substitution right away also if you look at sine to the fourth power if you want to break this apart s-send to a third power times central first power we are going to have a hard time to write sine to a third power in terms of cosine with ... Read More

Key Insights

• 😒 Integrating sine to the fourth power requires the use of u substitution and the power reduction formula for cosine.
• ✊ Breaking down sine to the fourth power into sine squared raised to the second power is a helpful strategy.
• 😑 Expanding the expression after applying the power reduction formula allows for easier integration.
• 🧑‍🏭 The constant multiples can be factored out of the integral to simplify the calculations.
• ✊ The power reduction formula for cosine involves transforming cosine to the first power and doubling the angle.

Questions & Answers

Q: How can we integrate sine to the fourth power?

To integrate sine to the fourth power, we rewrite it as sine squared raised to the second power and apply the power reduction formula to transform cosine to the first power.

Q: What is the power reduction formula for cosine?

The power reduction formula for cosine is 1/2 times (1 + cosine of 2 times the angle).

Q: How do we simplify the integral after applying the power reduction formula?

After applying the power reduction formula, we expand the expression and distribute the constants to simplify the integral into smaller components.

Q: What is the final answer for the integrated expression?

The final answer is 1/4 times (3/8 x - 1/4 sine of 2x + 1/32 sine of 4x) + C, where C is the constant of integration.

Summary & Key Takeaways

• The video explains how to integrate sine to the fourth power using u substitution and the power reduction formula.

• The strategy is to rewrite sine to the fourth power as sine squared raised to the second power and then apply the power reduction formula.

• The integral is simplified by expanding the expression and integrating each term individually.