Integral of (3x + 6)/sqrt(1 - x^2) | Summary and Q&A

TL;DR

The video demonstrates how to solve a challenging indefinite integral problem by employing the u-substitution and arc sine techniques.

Key Insights

• 😄 The initial attempt at solving the integral using u-substitution fails due to the presence of additional terms.
• 🥳 By breaking the integral into two parts, each part can be solved using different techniques.
• 👨‍💼 The first integral is solved using the u-substitution method, while the second integral is solved using the arc sine formula.
• ❓ The final solution involves combining the results obtained from solving both integrals.
• ❓ The integration process requires manipulating the equations to arrive at the correct solution.
• 😀 The plus C term represents the constant of integration and is included in the final solution.
• ☺️ The final solution includes the variables u, x, and the constant C.

Transcript

okay and this problem we're going to evaluate this indefinite integral this problem is a little bit harder because your initial suggestion your initial attempt when doing this problem would be to let u equals one minus x squared and at least mentally you should do this and then so d you will be negative 2x DX except you have a 3x plus 6 here so it ... Read More

Questions & Answers

Q: What is the initial attempt at solving the indefinite integral, and why does it not work?

The initial attempt is to let u equal 1 minus x squared. However, this approach fails because the presence of the term 6 complicates the substitution.

Q: How is the problem broken down to facilitate the solution?

The integral is split into two parts: 3x over the square root of 1 minus x squared and 6 over the square root of 1 minus x squared.

Q: What technique is used to solve the first integral?

The first integral is solved using the u-substitution method. By substituting u for the square root of 1 minus x squared, the integral becomes more manageable.

Q: How is the second integral solved?

The second integral can be solved using the arc sine formula. The integral of 1 over the square root of a squared minus x squared is equal to arc sine of x over a.

Summary & Key Takeaways

• The problem involves evaluating an indefinite integral that cannot be solved using the initial u-substitution attempt.

• To overcome the challenge, the problem is broken down into two separate integrals.

• The first integral is solved using the u-substitution method, while the second integral can be solved using the arc sine formula.