Integral of arctan(x^2)

TL;DR

This video demonstrates how to integrate the inverse tangent of x squared using a power series and find its radius of convergence.

Transcript

hi in this video we're going to integrate the inverse tangent of x squared with respect to x this is the same thing as the arc tangent and we're going to do it by evaluating it as a power series and we're also going to find the radius of convergence so first recall the formula that we usually use in these problems is a very basic one it says if you... Read More

Key Insights

• ☺️ The formula for 1 over 1 minus x can be used to represent various functions as infinite sums.
• 😑 Differentiating the inverse tangent function leads to an expression that can be represented as a power series.
• 👻 Integrating the power series allows us to obtain the integral of the inverse tangent function.
• ☺️ The radius of convergence for the power series representation of the inverse tangent function is determined by the absolute value of x being less than one.
• 📏 Understanding the chain rule is essential in finding the derivative of the inverse tangent function.
• ✊ Power series representations can simplify the process of evaluating integrals.
• ✊ The constant of integration introduced during integration can be represented by an additional constant term in the power series.

Questions & Answers

Q: What formula is used to represent 1 over 1 minus x when the absolute value of x is less than one?

The formula used is "1 over 1 minus x is equal to the infinite sum of x to the power of n, where n runs from zero to infinity."

Q: How is the derivative of the inverse tangent function found in this example?

The derivative of the inverse tangent function is found by differentiating the function 1 over 1 plus x squared squared, using the chain rule and the formula for the derivative of tangent inverse x.

Q: How is the power series representation of the derivative obtained?

The power series representation of the derivative is obtained by using the formula and simplifying the expression, resulting in an infinite sum of terms with alternating signs and increasing powers of x.

Q: How is the integral of inverse tangent x squared obtained from the power series representation?

The integral of inverse tangent x squared is obtained by integrating term by term and adding a constant of integration, resulting in an infinite sum of terms with increasing powers of x.

Summary & Key Takeaways

• The video explains the formula for 1 over 1 minus x and its infinite sum representation when the absolute value of x is less than one.

• It shows how to differentiate the inverse tangent function and create a power series representation.

• The process of integrating the power series term by term to obtain the integral of inverse tangent x squared is demonstrated.