integral of sqrt(tan(x))/sin(2x)

TL;DR

Learn how to integrate the square root of tangent x over sine 2x by simplifying the expression step by step.

Transcript

let's integrate square root of tangent x over sine of 2x here we have two travels first you see that we have tangent as citing an integral that's not what we would like okay usually would like to have sine and cosine in an integral and maybe we want to have tension in that case you want to have a secant okay sine cosine or tangent secant so I am go... Read More

Key Insights

• 😑 Integrating expressions with trigonometric functions often requires manipulations using trigonometric identities.
• 😑 Breaking down complex expressions into simpler forms can make it easier to integrate.
• 😑 Substitution is a useful technique to simplify expressions and make them easier to integrate.
• ❓ Understanding the properties and relationships of trigonometric functions is crucial in solving integration problems.

Questions & Answers

Q: How does the video suggest fixing the issue of having tangent in the integral instead of sine or cosine?

The video suggests fixing the issue by introducing the secant function, which is the reciprocal of cosine. By multiplying and dividing by the square root of cosine x, the expression is transformed using the identity sec^2 x = 1 + tan^2 x.

Q: Why does the video multiply the denominator by the square root of cosine x?

Multiplying the denominator by the square root of cosine x allows the video to convert the expression to a form where sine and cosine have the same angle. This simplification makes the integration process easier.

Q: How does the video simplify the expression further after the multiplication?

The video simplifies the expression by canceling out the square root of cosine x and bringing the secant squared x term to the numerator. This results in the expression 1/2 sec^2 x over the square root of tangent x.

Q: How does the video handle the square root of tangent x in the integral?

The video introduces a substitution by letting u = tangent x. By rewriting the integral in terms of u, the expression becomes 1/2 times the integral of u^(-1/2) du. This can be integrated easily using the power rule.

Summary & Key Takeaways

• The tutorial focuses on integrating the square root of tangent x over sine 2x by simplifying the expression.

• The video demonstrates how to break down the given expression and manipulate it to make the integration process easier.

• By using trigonometric identities and algebraic manipulation, the expression is simplified to a form that is easier to integrate.