Integral of 1/sqrt(3 - 4x - x^2)

TL;DR

Learn how to evaluate an indefinite integral by completing the square and using the arc sine formula.

Transcript

this problem we have to evaluate this indefinite integral so it looks like we might have to try to complete the square because if you just let you be this bottom piece here your D U is gonna have an accident and there's no X upstairs so let's start by trying to complete the square so we have 3 minus 4x minus x squared so let's rewrite this let's pu... Read More

Key Insights

• 😑 Completing the square is a technique that simplifies the integration process for quadratic expressions.
• 🤘 Adjusting the sign when completing the square is crucial for maintaining the equality.
• 🫠 The arc sine formula is a powerful tool for solving integrals involving a square root of a quadratic expression.
• ☺️ Making a suitable substitution, such as letting u be equal to x plus 2, can further simplify the integration process.
• ❓ It is important to be cautious when skipping steps in calculations to avoid potential errors.
• 🎮 The video emphasizes the importance of understanding each step, even if some may seem redundant.
• ❎ The square root of 7 is used as the coefficient in the arc sine formula after completing the square.

Questions & Answers

Q: What is the purpose of completing the square in this context?

Completing the square allows us to rewrite the quadratic expression in a way that makes it easier to integrate.

Q: How do you complete the square?

To complete the square, divide the coefficient of the linear term by 2, square the result, and add it to both sides of the equation.

Q: Why is there a need to adjust the sign when completing the square?

The sign adjustment is necessary because we initially multiplied the linear coefficient by -1. By adding and subtracting the same value, we maintain the equality.

Q: What is the formula used for integrating a term with a square root?

The arc sine formula, which states that the integral of dx over the square root of a squared minus x squared, is equal to arc sine of x over a.

Summary & Key Takeaways

• The video explains how to evaluate an indefinite integral by completing the square.

• Completing the square involves rearranging the terms to make the coefficient of the squared term 1.

• The arc sine formula is then used to solve the integral.