Integral of sin^3x

TL;DR

This lesson explains how to find the integral of sine cube X using substitution.

Transcript

in this lesson we're gonna talk about finding the integral of sine cube X DX now what we need to do is we need to expand this expression sine cube is basically sine squared times sine X DX now sine squared using the Pythagorean identity is 1 minus cosine squared now that expression comes from this equation sine squared plus cosine squared is equal ... Read More

Key Insights

• 👨‍💼 Expanding the expression of sine cube X into sine squared times sine X helps in simplifying the integral.
• ❎ The Pythagorean identity is used to express sine squared as 1 minus cosine squared.
• 👻 Substitution is a powerful technique in integration that allows us to transform the integral.
• ➖ The antiderivative of 1 minus u squared is u minus u to the third power divided by 3.
• 👨‍💼 Replacing the u variable with cosine X gives us the final answer for the integral of sine cube X.
• 😀 The constant C represents the arbitrary constant of integration.
• 🧊 The integral of sine cube X can be written as 1/3 cosine cube minus cosine X plus C.

Questions & Answers

Q: What is the Pythagorean identity and how is it used in finding the integral of sine cube X?

The Pythagorean identity states that sine squared plus cosine squared is equal to 1. By subtracting cosine squared from both sides, we can express sine squared as 1 minus cosine squared. This equation is then used to simplify the integral of sine cube X.

Q: Why do we use substitution in finding the integral of sine cube X?

Substitution allows us to transform the integral by replacing the variable, in this case, cosine X, with a new variable u. This simplifies the expression and makes it easier to integrate.

Q: How do we determine the antiderivative of 1 minus u squared?

The antiderivative of 1 is simply u, and the antiderivative of u squared is u to the third power divided by 3. Therefore, the antiderivative of 1 minus u squared is u minus u to the third power divided by 3.

Q: Why do we replace u with cosine X in the final answer?

In order to obtain the final answer for the integral of sine cube X, we need to replace u with cosine X. This is done because we made u equal to cosine X during the substitution process, and now we need to revert back to the original variable.

Summary & Key Takeaways

• The integral of sine cube X can be simplified by using the Pythagorean identity to express sine squared as 1 minus cosine squared.

• By making u equal to cosine X, the integral can be transformed into an expression with u.

• After integrating and replacing u with cosine X, the final answer for the integral of sine cube X is 1/3 cosine cube minus cosine X plus a constant.