Integral (arcsin(x))^2

TL;DR

Learn how to solve the integral of the arc sine of X squared using integration by parts, with step-by-step explanations.

Transcript

integrate arc sine of X quantity squared solution to do this we'll start by using integration by parts so recall the formula says if you have the integral of UDV that's equal to UV minus the integral of VDU so in this problem here we have the arc sine of x squared DX so we'll start by letting u be equal to arc sine of X quantity squared that forces... Read More

Key Insights

• 🥳 Integration by parts is a useful technique for solving integrals that involve products of functions.
• 🥳 The formula UV - integral of VDU is applied in integration by parts.
• 🍉 Identifying the appropriate U and DV terms is crucial for successful application of the method.
• 🥳 Sometimes, integration by parts needs to be used multiple times to simplify the integral.
• 🍉 Canceling out terms and performing substitutions can help simplify the integral further.
• 📏 The power rule and chain rule are essential tools for finding derivatives needed in integration by parts.
• 😑 Integrating an expression can involve applying various substitution techniques.

Questions & Answers

Q: What is the first step in solving the integral of the arc sine of X squared using integration by parts?

The first step is to identify U and DV. In this case, U is arc sine of X quantity squared, and DV is DX.

Q: How is DU calculated in this problem?

DU is calculated using the power rule, where the 2 is placed in front of arc sine of X to the first power. The derivative of the inside is obtained using the chain rule.

Q: How is V calculated?

V is calculated by integrating DV with respect to X, which gives X.

Q: Why is integration by parts used twice in this problem?

Integration by parts is used twice because the initial application results in an integral that can be simplified further using the same method.

Summary & Key Takeaways

• The video provides a step-by-step solution for finding the integral of the arc sine of X squared using integration by parts.

• The integration by parts formula (UV - integral of VDU) is applied to solve the problem.

• The solution involves using integration by parts twice, canceling out terms, and performing a U substitution.