Integral 1/(1 + sin(x)) from MIT Integration Bee Qualifying Exam 2018 Problem #9

TL;DR

The provided content explains the steps to solve the integral "integrity x over 1 plus sine x."

Transcript

integrity x over 1 plus sine x let's work through it solution this is from one of the MIT integration be qualifying exams I believe it's 2018 so the first thing you want to do is rewrite it and think of it as 1 over 1 plus sine X and it doesn't seem like there's an obvious use substitution it would be really nice if it was like some type of identit... Read More

Key Insights

• ❓ Rewriting the given integral and applying trigonometric identities can simplify the problem.
• 🔨 The difference of squares formula is a useful tool to manipulate the denominator and incorporate trigonometric identities.
• 🥺 Breaking down the integral into manageable terms can lead to further simplifications.
• 🆘 Recognizing functions and their derivatives can help in finding the integral solution.

Questions & Answers

Q: What is the first step in solving the integral "integrity x over 1 plus sine x"?

The first step is to rewrite the integral as "1 over 1 plus sine x" and consider using trigonometric identities.

Q: How is the difference of squares formula used in the solution?

By multiplying the numerator and denominator of "1 minus sine x" by "1 minus sine x," the difference of squares formula is applied to simplify the denominator.

Q: How does the integral simplify after applying the trigonometric identity and the difference of squares formula?

The integral becomes "1 over cosine squared x" minus "sine x over cosine squared x," which can be further re-written as "tangent x secant x" using trigonometric functions.

Q: What functions have derivatives that match the simplified integral terms?

The derivative of "tangent x" is "secant squared x," and the derivative of "secant x" is "secant x tangent x," which helps in integrating the terms.

Summary & Key Takeaways

• The content provides a step-by-step solution for integrating "integrity x over 1 plus sine x."

• It suggests rewriting the integral as "1 over 1 plus sine x" and using a trigonometric identity.

• The difference of squares formula is used to simplify the integral, leading to the final solution.