Integral of (1+x)/(1+x^2), tricky u-substitution problem, calculus 1 tutorial

TL;DR

Break down a complex integral into two fractions, solve individually, and combine for the final answer.

Transcript

five point five number 45 we are going to integrate 1 plus x over 1 plus x squared this is one of the first tricky questions that you are going to encounter with the integral but let me show you how to do it where the thing right here is that if you just want to use a u substitution right here right away then it's not going to work if you let u is ... Read More

Key Insights

• 🍳 Integrating complex fractions involves breaking them down into simpler components.
• ☺️ The integral of 1 plus x over 1 plus x squared requires handling two separate fractions.
• ☺️ The inverse tangent function is used to solve the integral of 1 over 1 plus x squared DX.
• ☺️ A u-substitution is employed to solve the integral of x over 1 plus x squared DX.
• 🥺 Properly handling the components and following the correct steps leads to the accurate integration result.
• ❓ The final answer combines the individual integrals of the separated fractions.
• ❓ Understanding the nature of the function enables selecting the appropriate integration method.

Questions & Answers

Q: How can the integral 1 plus x over 1 plus x squared be solved?

To solve this integral, break the fraction into two parts - 1 over 1 plus x squared and x over 1 plus x squared DX. Solve each part individually and combine the results for the final answer.

Q: What function is used to solve the integral 1 over 1 plus x squared DX?

The function used to solve the integral 1 over 1 plus x squared DX is the inverse tangent function, yielding an inverse tangent X as the result.

Q: How is the integral of x over 1 plus x squared DX solved?

The integral of x over 1 plus x squared DX is solved using a u-substitution. Let u be the denominator 1 plus x squared, and after substitution and calculation, the result is 1/2 Ln of 1 plus x squared.

Q: Why is it important to break down the fraction before integrating?

Breaking down the fraction into two separate fractions makes it easier to solve the integral by focusing on each part individually, leading to a simpler and more manageable calculation process.

Summary & Key Takeaways

• Integrating 1 plus x over 1 plus x squared involves breaking the fraction into two parts: 1 over 1 plus x squared and x over 1 plus x squared DX.

• For the first part, the integral is solved using the inverse tangent function, while the second part requires a u-substitution.

• The final answer is the sum of the integrals of the two fractions, yielding inverse tangent X plus 1/2 Ln of 1 plus x squared + C.