integral of (sin^-1(x))^2, integration by parts

TL;DR

Learn how to use the DI method to integrate the square of arcsin(x) step by step.

Transcript

let's do integration by parts for the integral  of (arcsin(x))^2 and let me show you with the   DI method we are going to be encountering string  things that's okay let me show you how it goes   anyways for the di method it says i'm going to  pick something to be differentiated and then   something to be integrated and we only have one  thing right... Read More

Key Insights

• 🥳 Integration by parts involves differentiating and integrating two parts of a function.
• ✋ Identifying when to stop during the integration process is crucial to ensure the integral remains manageable.
• 🫤 The product of the diagonal obtained from the DI method gives the final answer to the integral.

Questions & Answers

Q: How do you start the integration by parts for the integral of (arcsin(x))^2?

To start, you apply the DI method by differentiating "arc sin x^2" and integrating the constant term 1. This is done to simplify the integral before proceeding further with the calculations.

Q: Why do we need to stop at a certain point during the integration process?

It is necessary to stop at a certain point because continuing the integration would result in an increasingly larger expression. Additionally, the product of the expressions obtained at each step allows us to find the final integral, so further steps are not required.

Q: What is the product of the diagonal and how does it lead to the final answer?

The product of the diagonal is the answer to the integral. In this case, the product is obtained by multiplying "x" with "arc sin x^2". Each row in the diagonal represents a different integral, and when combined, they give the complete solution.

Q: Why does the DI method alone not work for some integrals?

The DI method does not work for every integral due to the complexity of certain functions. This is the reason why, in some cases, additional methods like u substitution or integration by parts are needed to solve specific parts of the integral.

Summary & Key Takeaways

• This content explains how to integrate the square of arcsine using the DI (Differentiate and Integrate) method.

• The process involves differentiating one part and integrating the other part, and recognizing when to stop.

• By applying integration by parts multiple times and using u substitution, the integral is solved and the final answer is obtained.