Integral sech(x) from the MIT Integration Bee Qualifying Exam 2017 Problem #3

TL;DR

Integrate hyperbolic secant of X using clever substitutions and trigonometric identities to solve the integration problem.

Transcript

integrate the hyperbolic secant of X let's try it solution so this problem is from one of the MIT integration be qualifying exams so I am thinking that we know that the hyperbolic secant of X is 1 over the hyperbolic cosine of X and so we know that the hyperbolic cosine of X that's the average of e to the X and e to the negative x so you add them u... Read More

Key Insights

• ❓ Understanding the relationship between hyperbolic functions can simplify integration problems.
• 🥺 Clever manipulations and substitutions can lead to elegant solutions in calculus.

Questions & Answers

Q: What is hyperbolic secant of X and its relationship with hyperbolic cosine of X?

Hyperbolic secant of X is 1 over hyperbolic cosine of X, which is the average of e to the X and e to the negative x.

Q: How is the integral of hyperbolic secant of X solved step by step?

The solution involves cleverly multiplying and manipulating terms to simplify the integral, followed by a substitution using the arc tangent formula.

Summary & Key Takeaways

• Explanation of hyperbolic secant of X being 1 over hyperbolic cosine of X.

• Clever manipulation by multiplying to get rid of e to the negative x.

• Substituting e to the X as u to apply the arc tangent formula.