integral of sin^-1(x) | Summary and Q&A

TL;DR

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

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).

Transcript

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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.