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.

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.

Transcript

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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.