Integral (1 + e^x)/(1 - e^x) from MIT Integration Been Qualifying Exam 2015 Problem #7

TL;DR

Using a clever trick to simplify integrals involving e to the x in the numerator and denominator.

Transcript

integrate 1 plus e to the x over 1 minus e to the X solution so usually in these problems where you have EES in the numerator and/or the denominator you can try to make some type of substitution or oftentimes there's a clever trick let's do this using a clever trick so we have 1 plus e to the x over 1 minus e to the X DX so we can write this as the... Read More

Key Insights

• 🍉 Adding and subtracting terms cleverly can simplify complex integrals involving exponential functions.
• 🥳 Breaking up integrals into simpler parts can lead to easier integration and solution of the overall problem.
• 💁 Using substitutions can transform integrals into more manageable forms for easier calculation.
• ❓ Clever strategies and techniques are essential in effectively solving challenging mathematical problems.
• ⌛ Practice with integration techniques, like the one demonstrated, can help maintain and improve mathematical skills over time.
• ❓ Understanding the principles behind integration can enhance problem-solving abilities in more complex mathematical scenarios.
• ❓ Clever tricks and strategies in mathematics can streamline the process of solving intricate problems efficiently.

Questions & Answers

Q: How can you simplify integrals involving e to the x in the numerator and denominator?

By cleverly adding and subtracting e to the x in a strategic manner, you can cancel out terms and simplify the integral, making it more manageable and easier to solve.

Q: What is the significance of breaking up the integral into two parts in this problem?

Breaking up the integral allows for one part to be easily integrated to 1 while the other part involves a logarithmic function, simplifying the overall integration process.

Q: How does making a substitution with u as 1 minus e to the x help in solving the integral?

By substituting u for 1 minus e to the x, the integration becomes easier as it transforms into a simpler form that can be easily solved using basic integration techniques.

Q: Why is this clever integration technique useful in solving complex integrals?

This technique allows for the simplification of integrals involving exponential functions, making them more accessible and manageable for integration, showcasing the importance of clever strategies in mathematical problem-solving.

Summary & Key Takeaways

• Integrating 1 plus e to the x over 1 minus e to the x can be simplified by adding and subtracting e to the x cleverly.

• By breaking up the expression into two integrals, one part simplifies to the integral of 1 while the other involves a simple logarithm.

• Making a substitution with u being 1 minus e to the x helps in simplifying the integration process.