improper integral of x*e^(-x^2) from -inf to +inf

TL;DR

Breaking apart the integral from negative infinity to positive infinity into two parts allows us to solve it; the final answer is zero.

Transcript

another improper integral the integral from negative Infinity to positive infinity x * e to x² unfortunately we see both Infinities right here the negative and the positive infinity and remember we can only deal with one Infinity at a time in the integral so we have to break this apart into two pieces so this is what we can do for the first integra... Read More

Key Insights

• 🍳 An improper integral with infinity as the limits must be broken down into two separate integrals.
• 😥 Choosing the same finite values for the starting and ending points in each integral is permissible.
• 🆘 Calculating the antiderivative for each part helps determine if the integral converges or diverges.
• 🥳 In this case, the antiderivative for both parts resulted in finite values, indicating convergence.
• ♾️ The final answer for the improper integral from negative infinity to positive infinity is zero.
• 🍳 Understanding the concept of breaking apart an improper integral is important for solving similar problems in calculus.
• 🥳 The choice of the starting and ending values for the integrals can vary as long as they are finite and the same for both parts.

Questions & Answers

Q: Why do we need to break the improper integral into two parts?

We need to break the integral into two parts because we can only deal with one infinity at a time. By splitting it into two, we can evaluate each part individually.

Q: What values can we choose for the starting and ending points in each integral?

As long as the values are finite and the same for both integrals, any number can be chosen. In this case, zero was chosen for simplicity.

Q: How do we determine if the integral converges or diverges?

To determine if the integral converges or diverges, we evaluate the antiderivative for each part of the integral. If a finite value is obtained, the integral converges; otherwise, it diverges.

Q: What is the final answer for the improper integral?

The final answer for the improper integral is zero. This means that the integral converges.

Summary & Key Takeaways

• The given content explains how to solve an improper integral from negative infinity to positive infinity by breaking it into two separate integrals.

• To do this, we choose a finite number for the starting and ending points in each integral.

• The antiderivative of the integrand is calculated for both parts, which yields the final answer of zero.