Q54, Integral of sqrt(x^2+4x+5), trig sub with completing the square

TL;DR

Learn how to integrate the square root of a quadratic equation with three terms using the completing the square method.

Transcript

okay I heard that you are taking Kyle to therefore have this right here for you let's take a look at how we can integrate the square root of x squared plus 4x plus 5 you know usually when you have a quadratic in set up a square root we just want to have two terms so you can do tricks up but here we have three terms it's not that bad at all because ... Read More

Key Insights

• ❎ Integrating the square root of a quadratic equation with three terms can be done using the completing the square method.
• ❎ Completing the square involves finding the "magic number" by adding 1/2 times the coefficient of x squared and squaring it.
• 💯 The quadratic equation is then rewritten as a perfect square, which can be factored and simplified before integration.
• 😑 The tangent function can be used to simplify the expression inside the square root.
• 🥳 Integration by parts is necessary to integrate secant cubed theta.

Questions & Answers

Q: How do you integrate the square root of a quadratic equation with three terms?

To integrate the square root of a quadratic equation with three terms, you can use the completing the square method. This involves finding the "magic number" by adding 1/2 times the coefficient of x squared and squaring it.

Q: What is the purpose of completing the square?

Completing the square allows us to rewrite the quadratic equation as a perfect square, which can be factored and simplified before integration. This makes the integration process easier.

Q: How do you find the "magic number" in completing the square?

To find the "magic number," add 1/2 times the coefficient of x squared and square it. This number is used to make the first three terms of the quadratic equation a perfect square.

Q: Do we need to use absolute value in the final answer?

No, the absolute value is not needed in the final answer. Since the expression inside the square root is always positive, there is no need to include the absolute value.

Summary & Key Takeaways

• The video explains how to integrate the square root of a quadratic equation with three terms using completing the square.

• Completing the square involves finding the "magic number" by adding 1/2 times the coefficient of x squared and squaring it.

• The quadratic equation is then rewritten as a perfect square, which can be factored and simplified before integration.