Q75, integral of 1/(1+x^2)^(5/2), trig sub

TL;DR

Learn how to integrate 1/(1+x^2)^(5/2) by using trigonometric substitution and simplifying the expression step by step.

Transcript

okay insidious show you guys how to integrate 1 over 1 plus x squared in a parenthesis raised to the 5 over 2 power in this case unfortunately we don't have the X on the top otherwise we can just let u equal to 1 plus x squared but don't worry because we could just do tricks up right here especially we notice that we have this power 5 over 2 notice... Read More

Key Insights

• ❓ Trigonometric substitution is a useful technique for simplifying integrales involving rational functions.
• 😑 By substituting x with tan(theta), the expression can be rewritten in terms of trigonometric functions.
• 👻 Applying trigonometric identities allows for further simplification.

Questions & Answers

Q: How does the video approach the integration problem?

The video utilizes trigonometric substitution by substituting x with tan(theta) to simplify the expression and make it easier to integrate.

Q: Why does the video use the power of 5/2 for the denominator?

The power of 5/2 is used in the denominator because it allows for simplification of the expression by utilizing trigonometric identities.

Q: What is the final solution to the integral of 1/(1+x^2)^(5/2)?

The final solution is X/(sqrt(1+X^2))^(3/2) + C, where C represents the constant of integration.

Q: Can the same approach be used for other similar integrals?

Yes, the same approach of trigonometric substitution can be used for other integrals involving expressions with a quadratic term.

Summary & Key Takeaways

• The video explains how to integrate the expression 1/(1+x^2)^(5/2) using trigonometric substitution.

• By substituting x with tan(theta) and manipulating the expression, the integral transforms into integrating 1-(sin^2(theta))^3, making it easier to solve.

• After simplifying and converting back to the x variable, the integral solution is derived as X/(sqrt(1+X^2))^(3/2) + C.