Integral of 1/sqrt(4 + x^2) : Example Involving the Inverse Hyperbolic Sine Function

TL;DR

Learn how to evaluate an indefinite integral using a specific formula and u-substitution.

Transcript

hi everyone in this problem we have to evaluate this indefinite integral so the formula that we're going to try to use in this problem is the following if you have the integral of d u over the square root of one plus u squared this is equal to cinch inverse of u plus our constant and of integration capital c so this is the formula that we're going ... Read More

Key Insights

• ❓ Understanding and applying specific formulas can simplify integral evaluations.
• 😑 Manipulating expressions to fit the formula requirements is crucial in solving mathematical problems.
• 🥋 U-substitution can help transform complex integrals into simpler forms for easier evaluation.
• 💄 Recognizing patterns and making appropriate substitutions can enhance problem-solving efficiency in mathematics.

Questions & Answers

Q: What formula is used to evaluate the indefinite integral in the video?

The formula utilized is the integral of du over the square root of 1+u^2, which corresponds to cinh inverse of u plus a constant of integration.

Q: How is the given problem manipulated to fit the formula requirements?

The problem is altered by expressing square root of 4+x^2 as 1+x^2/4 then simplified into 1+x^2/4 as x/2 squared, allowing for u-substitution.

Q: What is the u-substitution made in the problem, and how is it implemented?

The u-substitution is u = x/2, leading to du = 1/2 dx, aligning with the required formula and simplifying the integral evaluation process.

Q: What is the final solution of the indefinite integral in the video?

The solution is cinh inverse of x/2 plus a constant of integration, providing the evaluated indefinite integral result.

Summary & Key Takeaways

• Utilize a formula involving the integral of du over the square root of 1+u^2 to evaluate an indefinite integral.

• Manipulate the given problem to match the formula by adjusting the expression with u-substitution.

• Apply the formula after making the u-substitution to find the indefinite integral solution.