How to Integrate 1/(5x² - 30x + 65)

TL;DR

To solve the integral of 1/(5x² - 30x + 65), first complete the square in the denominator to rewrite the expression. Then, use u substitution to simplify and find the integral, resulting in (1/10) * arctan((x - 3)/2) + C.

Transcript

• [Instructor] All right, let's see if we can find the indefinite integral of one over five x squared minus 30x plus 65 dx. Pause this video and see if you can figure it out. All right, so this is going to be an interesting one. And it'll be a little bit hairy, but we're gonna work through it together. So, immediately you might try multiple integra... Read More

Key Insights

• ❎ Completing the square in the denominator simplifies the integration process.
• 🆘 U substitution helps to connect the integral with the derivative of arctan.
• 🧑‍🏭 Factoring out common terms and simplifying fractions can make the solution more manageable.

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.