Help with improper integral calculation

In summary, the conversation is about finding the integral of a function involving e^x and using techniques such as u-substitution and integration by parts. The solution involves setting u=e^(5x) and using the fact that dx=du/(5u) to simplify the integral, which can then be solved using previous knowledge.
  • #1
I'm supposed to find the integral of f(x) = (e^5x) / (1+(e^10x)) from negative infinity to 0. I know how to set up the integral as the limit as t approaches -∞ of ∫f(x) from t to 0, but I'm stuck on how to actually solve the integral. I've tried by parts and u-sub but I just can't seem to get it.
Any help would be appreciated, thanks in advance!
 
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  • #2
Well, if you set u=e^(5x), u^2=e^(10x).
Furthermore, you have:
du/dx=5u, that is dx=du/(5u)
Negative infinity in "x" goes to 0 in "u", whereas 0 in "x" goes to 1 in "u"
Your new integrand becomes 1/5*(1/(1+u^2)), which you should know how to integrate.
 

What is an improper integral?

An improper integral is an integral where one or both of the limits of integration are infinite or where the integrand is unbounded at one or more points in the interval of integration. These integrals do not have a finite value and require special techniques to evaluate.

What is the difference between a convergent and divergent improper integral?

A convergent improper integral is one where the limit of the integral exists and has a finite value. A divergent improper integral is one where the limit of the integral does not exist or has an infinite value.

How do I determine if an improper integral is convergent or divergent?

To determine if an improper integral is convergent or divergent, you can use a variety of tests such as the comparison test, limit comparison test, or the integral test. These tests compare the improper integral to a known convergent or divergent series.

What are some common techniques for evaluating improper integrals?

Some common techniques for evaluating improper integrals include using limits, changing variables, and using integration by parts. These techniques can help transform the integral into a form that can be evaluated using basic integration rules.

Are there any special cases of improper integrals?

Yes, there are two special cases of improper integrals: Type I and Type II. Type I improper integrals have a finite interval of integration, but one or both of the limits are infinite. Type II improper integrals have a finite interval of integration, but the integrand is unbounded at one or more points in the interval.

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