improper integral of x*e^(-x^2) from -inf to +inf | Summary and Q&A

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February 25, 2015
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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.

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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.

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