integral of arcsin(x)

TL;DR

The integral of arc sine of x is solved using integration by parts, resulting in x arc sine of x plus the square root of 1 minus x squared.

Transcript

what is the integral of arc sine of x or the inverse sine of x dx how can we begin this problem what we need to use is integration by parts so we're going to use the formula the integral of u dv is uv minus the integral of v d u so what we're going to do is we're going to make u equal to sine x or arcsin x rather and dv is going to be dx so if dv i... Read More

Key Insights

• 🥳 Integration by parts is a technique used to solve certain types of integrals.
• 🗂️ The derivative of arc sine of x is 1 divided by the square root of 1 minus x squared.
• 🎓 Applying the integration by parts formula involves choosing u and dv, finding du and v, and plugging them into the formula.
• 😑 Simplification of the integral expression is important to ensure the answer is in the simplest form.
• 🆘 Using u substitution can help simplify the integral and make it easier to solve.
• 😑 The final expression for the integral of arc sine of x includes x arc sine of x and the square root of 1 minus x squared.
• 🫠 Understanding the derivative of arc sine of x is crucial in solving the integral.

Questions & Answers

Q: How is the integral of arc sine of x approached in this problem?

The integral is solved using integration by parts, where u is set as arc sine of x and dv is dx.

Q: What is the derivative of arc sine of x?

The derivative of arc sine of x is 1 divided by the square root of 1 minus x squared.

Q: What is the final expression for the integral of arc sine of x?

The final expression is x arc sine of x plus the square root of 1 minus x squared.

Q: Can the answer be simplified further?

No, the expression x arc sine of x plus the square root of 1 minus x squared is the simplest form of the integral.

Summary & Key Takeaways

• The integral of arc sine of x is solved using integration by parts.

• The derivative of arc sine of x is 1 divided by the square root of 1 minus x squared.

• Applying the integration by parts formula, the final answer is x arc sine of x plus the square root of 1 minus x squared.