Trigonometric Integrals Powers of Secant and Tangent sec^4(2x)*tan^4(2x)  Summary and Q&A
TL;DR
Learn how to integrate even powers of secant and tangent using trigonometric identities and substitution.
Key Insights
 ✊ Integrating powers of secant and tangent involves saving a copy of secant squared and using trigonometric identities.
 💄 Making the appropriate substitution, such as u = tan(x), simplifies the expression and helps in obtaining the antiderivative.
 ❤️🩹 Plugging in the original trigonometric function values at the end of the integration process gives the complete solution.
 ✊ Even powers of secant and tangent can be integrated using the same approach, while odd powers require saving a copy of secant multiplied by tangent.
Transcript
you have to integrate secant to the fourth power of two x times the tangent to the fourth power of two x solution so we have an even secant and even power of secant so what we're gonna do is we're gonna save a copy of secant squared 2x right so whenever you have an even power of secant this usually works now if you have an odd power of tangent you ... Read More
Questions & Answers
Q: How can I integrate even powers of secant and tangent?
You can integrate even powers of secant and tangent by saving a copy of sec^2(2x) and using the identity sec^2(2x) = 1 + tan^2(2x). Then, make the substitution u = tan(x) and simplify the expression.
Q: What should I do if I have an odd power of tangent in the integral?
If you have an odd power of tangent, you would save a copy of secant multiplied by tangent. The rest of the process remains the same.
Q: What is the purpose of making the substitution u = tan(x)?
Making the substitution u = tan(x) allows you to rewrite the integral in terms of u, simplifying the expression and eliminating the trigonometric functions. This makes the integration process easier.
Q: How do I obtain the final solution after simplifying the expression?
After simplifying the expression in terms of u, you can integrate it in terms of u. Then, replace u with tan(2x) to obtain the final solution of the integral.
Summary & Key Takeaways

To integrate even powers of secant and an even number of powers of tangent, save a copy of sec^2(2x) and rewrite the expression using the identity sec^2(2x) = 1 + tan^2(2x).

Make the substitution u = tan(x) and rewrite the integral in terms of u, simplifying the expression and eliminating the trigonometric functions.

Integrate the simplified expression in terms of u and replace u with tan(2x) to obtain the final solution.