Q60, integral of 1/(x^2*sqrt(1-x^2)), trig sub vs u sub

TL;DR

This video demonstrates how to integrate the expression 1/(x^2 * sqrt(1-x^2)) using trigonometric substitution.

Transcript

okay in this video show us how to integrate 1 over x squared x squared root of 1 minus x squared so as we can see we don't have any egg RPS juice up right it's not x over square root of 1 minus x squared it's not like that so for this kind of situations let's go ahead and use Twitter for this and because the inside here we have 1 minus x squared I'... Read More

Key Insights

• 😑 Trigonometric substitution is a useful technique for integrating expressions that do not have a simple antiderivative.
• 😑 Choosing the right substitution can greatly simplify the expression and make the integration process easier.
• 😑 After integrating the expression in terms of the new variable, the solution can be converted back to the original variable using trigonometric identities.

Summary & Key Takeaways

• The video explains how to integrate the given expression by using trigonometric substitution.

• The speaker shows two possible substitutions: X=sin(theta) and X=cos(theta), and chooses X=sin(theta) for this specific problem.

• By substituting X=sin(theta) and simplifying the expression, the integral can be converted into an integral of 1/sin^2(theta), which can be easily solved.

• Finally, the speaker converts the solution back into the original variable, X, and presents the final answer.