Integration using completing the square and the derivative of arctan(x)  Khan Academy  Summary and Q&A
TL;DR
Learn how to solve the indefinite integral of 1/(5x^2  30x + 65) and simplify it to the form of arctan.
Questions & Answers
Q: How is the integral of 1/(5x^2  30x + 65) solved?
The process involves completing the square in the denominator, which simplifies it into the form of the derivative of arctan. Then, with the help of u substitution, the integral is further simplified to (1/10) * arctan((x  3)/2) + C.
Q: Why is completing the square necessary in this integral?
Completing the square is necessary to simplify the denominator into the form of a perfect square, which makes it easier to integrate. It allows us to recognize the connection to arctan and simplifies the solution process.
Q: What is the purpose of u substitution in this integral?
U substitution is used to further simplify the integral by letting u = (x  3)/2. This substitution allows for a direct connection to the derivative of arctan, making the overall integration process easier.
Q: How can the integral be simplified even more after u substitution?
After u substitution, the integral becomes (1/10) * arctan(u) + C, where u = (x  3)/2. This is the simplest form of the integral and cannot be further simplified.
Summary & Key Takeaways

The video explains how to find the indefinite integral of 1/(5x^2  30x + 65).

The process involves completing the square in the denominator to simplify it into the form of the derivative of arctan.

By implementing u substitution, the integral is further simplified and yields the final result of (1/10) * arctan((x  3)/2) + C.