How to Integrate csc(x)sec(x) Using Trigonometric Identities

TL;DR

To integrate cosecant x times secant x, rewrite it as 1 over sine x times 1 over cosine x. Use the identity sin(2x) = 2sin(x)cos(x) to simplify the integral, which leads to 2 times the integral of cosecant(2x) dx. The final result utilizes the formula for the integral of cosecant, yielding -ln|csc(2x) + cot(2x)| + C.

Transcript

hey what's up YouTube this problem I'm going to integrate cosecant of X times the secant of X with respect to X solution so when you first see this problem you probably think is there a u-substitution I wasn't able to find one if you let u be secant and the derivative is secant tangent that doesn't work if you let you be cosecant the derivative is ... Read More

Key Insights

• 💦 Traditional u-substitution does not work for every integration problem.
• 😒 Rewriting trigonometric functions can allow for the use of identities and simplification.
• ❓ The integral of cosecant has a formula that can be modified for different variations of the function.

Questions & Answers

Q: Why doesn't traditional u-substitution work for this problem?

Traditional u-substitution relies on finding a substitution where the derivative matches part of the integrand. However, neither the derivative of secant nor the derivative of cosecant produces a suitable substitution.

Q: How does rewriting cosecant and secant help in this problem?

By rewriting cosecant as 1 over sine x and secant as 1 over cosine x, the integral is transformed into 1 over sine 2x. This simplification allows for the application of the sine 2x identity.

Q: What is the formula for the integral of cosecant?

The formula is -ln |cosecant x| + cotangent x + C, where C is a constant. In the given problem, the formula is modified to account for the 2x term by using a u-substitution.

Q: Can this problem be solved using other methods?

Yes, there are multiple approaches to integrating the given function. This video presents one particular approach, but there may be alternative methods that yield the same result.

Summary & Key Takeaways

• The video explains that traditional u-substitution doesn't work for the given problem.

• The solution involves rewriting cosecant as 1 over sine x and secant as 1 over cosine x.

• The identity sine 2x = 2sin x cos x is used to simplify the integral, which then matches the formula for the integral of cosecant.