How do I compute the integral of x/(e^x) with limits 0 and infinity?

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In summary, the conversation discusses computing an integral that converges, specifically the integral of x/(e^x) with lower limit 0 and upper limit infinite. The individual asking for help figures out the solution by using integration by parts and applying l'Hôpital's rule, resulting in the answer of 1. Additionally, there is mention of the gamma function and its relation to the integral form.
  • #1
steelphoenix
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Im doing a chapter on Improper integrals, and one of my problems is i need to compute an integral if it converges. I really don't know where to start with this one. Anyone point me in the right direction?

integral of x/(e^x) where the lower limit is 0 and the upper is infinite

really any help would be appreciated, I got a bit of a cold and seem to be forgetting every bit of math
 
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  • #2
ook wait, i think i figured it out

i stat off by calculating the integral, which turns out to be

-( (x+1)/e^x )

then you end up getting 1 after you plug in the limits, because when its infinity it becomes zero and when 0 it becomes -1, so when you subtract negetive 1 it becomes 1, and 1 is the answer
 
  • #3
You have x/(e^x) = [tex]xe^{-x}[/tex]

So [tex]\int_{0}^{\infty}xe^{-x}dx[/tex]

Integration by parts gives [tex]{\lim }\limits_{x \to \infty } -e^{-x}(x+1) + 1[/tex]

[tex]{\lim }\limits_{x \to \infty } \frac{-(x+1) }{e^x} + 1[/tex]

Applying l'Hôpital's rule gives [tex]{\lim }\limits_{x \to \infty } \frac{-1 }{e^x} + 1[/tex] = 0 + 1
= 1
 
  • #4
As an aside, in case anyone cares:

[tex]\int_0^\infty x^n e^{-x}\,dx = n![/tex]

For integer n. However, the integral form is also well-defined for real, and even complex n, except at the negative integers where it diverges. Look up "gamma function" for more details.
 
  • #5
I somehow recognized it before I read the rest of your post. Math is fascinating
 

Related to How do I compute the integral of x/(e^x) with limits 0 and infinity?

1. What is an E integral?

An E integral is a type of mathematical integral used in quantum mechanics to calculate the energy of a system. It is often used to solve the Schrödinger equation, which describes the behavior of quantum particles.

2. How do I solve an E integral?

To solve an E integral, you will need to use integration techniques such as substitution or integration by parts. You will also need to have a good understanding of mathematical concepts such as derivatives and definite integrals.

3. What is the significance of the E integral in quantum mechanics?

The E integral plays a crucial role in quantum mechanics as it allows scientists to calculate the energy of a quantum system. This information is essential in understanding the behavior and properties of particles at a microscopic level.

4. Are there any specific conditions for solving an E integral?

Yes, there are certain conditions that need to be met in order to solve an E integral accurately. These include having a well-defined potential function, knowing the boundary conditions, and using the appropriate mathematical techniques.

5. Can I use a computer to solve an E integral?

Yes, it is possible to use a computer to solve an E integral. There are various software programs and online tools available that can help with the calculations. However, it is still important to have a good understanding of the concepts and techniques involved in solving the integral.

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