integral of sin^-1(x)

TL;DR

In order to integrate the inverse sine function, we can use integration by parts by choosing appropriate functions for u and dv.

Transcript

let's integrate the inverse sine function but then it seems that we don't have a lot of things to work with and I don't know the root of what what give me in for saying is right away right however we can still use integration by parts for this question and let me show you for integration by parts once again we have to choose something for you and t... Read More

Key Insights

• 🛫 Integration by parts is a useful technique for evaluating integrals by choosing appropriate functions for u and dv.
• 👨‍💼 In this specific example, the inverse sine function is chosen as u and dx as dv.
• 🧑 The derivative of the inverse sine function is 1/sqrt(1-x^2), which is used to find du.
• ☺️ The integral of dx is simply x, which is used to find v.
• 🥳 After applying the formula for integration by parts, a substitution is used to evaluate the remaining integral.
• ☺️ The final result is the integral of x times the inverse sine of x minus the integral of sqrt(1-x^2).
• ✊ The integral of sqrt(1-x^2) can be evaluated using a reverse power rule and simplifies to -sqrt(1-x^2).
• ☺️ The resulting integral is then simplified to x times the inverse sine of x plus sqrt(1-x^2).

Questions & Answers

Q: What is integration by parts?

Integration by parts is a technique used to evaluate integrals by choosing appropriate functions for u and dv, and then applying the formula that involves multiplying u and v together.

Q: How do we choose the functions u and dv for integration by parts?

The choice of u and dv depends on the integrand. We generally choose u as a function that becomes simpler after differentiation, and dv as a function that can be easily integrated.

Q: How do we integrate the inverse sine function using integration by parts?

To integrate the inverse sine function, we choose u as the inverse sine of x and dv as dx. We differentiate u to find du and integrate dv to find v.

Q: What do we do after choosing u and dv for the inverse sine function?

After choosing u and dv, we apply the formula for integration by parts, which involves multiplying u and v together and subtracting the integral of vdu.

Summary & Key Takeaways

• Integration by parts is a method used to evaluate integrals by selecting functions for u and dv in order to simplify the integral.

• To integrate the inverse sine function, we choose u as the inverse sine function, and dv as dx.

• We differentiate u to find du, and integrate dv to find v.

• Using the formula for integration by parts, we multiply u and v together and subtract the integral of vdu.

• The resulting integral is then simplified using a substitution to evaluate the remaining integral.