integral of sqrt(x+sqrt(x+sqrt(x+...))), infinite nested square root

TL;DR

Learn how to integrate functions with square roots using the method of completing the square.

Transcript

okay last time showed you guys how to differentiate square root of x plus square root of x plus dot dot this time show you guys how to integrate this guy so first let me just do a quick review with you guys I will begin by saying there y equal to this we have infinitely many square root of x plus square root of x plus square root of x plus dot dot ... Read More

Key Insights

• ❓ Recognizing the connection between the original function and the integrated function simplifies the process of integration.
• 🥘 Completing the square allows us to rewrite the function in terms of a variable y, making it easier to integrate.
• 😑 Considering the sign of the square root expression is essential to ensure the resulting function is always positive.
• 😑 Integrating the square root function involves factoring out a constant and performing the integral of the remaining expression.
• ❎ The process of completing the square involves adding a specific constant term to both sides of the equation to create a perfect square.
• 😑 The use of substitution and the power rule are applied when integrating the remaining expression after completing the square.
• 🫚 The result of the integration is a constant multiple of the integral of the square root function, plus an arbitrary constant.

Questions & Answers

Q: What is the connection between the original function and the integrated function?

The integrated function is obtained by recognizing that the original function can be written as the square root of x plus the integrated function itself. This allows us to rewrite the original function in terms of a variable y.

Q: How do we complete the square in the process of integration?

To complete the square, first make sure the coefficient of the squared term is 1. Next, take half of the coefficient of the linear term, square it, and add it to both sides of the equation. This guarantees that the left-hand side becomes a perfect square.

Q: Why is it important to consider the sign of the square root expression in the final result?

It is crucial to consider the sign of the square root expression because we want to ensure that the resulting expression is always positive. This is necessary to avoid obtaining negative results when evaluating the integrated function for certain values of x.

Q: How do we integrate the square root function after completing the square?

After completing the square, we can integrate the function by expressing it as a constant multiple of a standard integral. The constant multiple is factored out, and the integral of the remaining expression is evaluated using the rules of integration.

Summary & Key Takeaways

• The content teaches how to integrate square root functions by completing the square.

• By recognizing the connection between the original function and the integrated function, the process becomes easier.

• The integration involves factoring out a constant and performing the integral of the remaining expression.