Integrate x^(1/2)*e^(-x): Tips & Tricks

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In summary, the conversation discusses a difficult integral involving the function x^(1/2) multiplied by e^(-x), with integration limits from -infinity to infinity. Various methods have been suggested, including integrating by parts and substituting x=t^2, but it is noted that the function is not defined on one of the intervals. The idea of breaking the integral into two parts is also mentioned, but it is acknowledged that the function may still diverge.
  • #1
snoweangel27
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I can't seem to figure this integral out
[tex]\int[/tex]x^(1/2)*e^(-x) dx
from -[tex]\infty[/tex] to [tex]\infty[/tex]

I have tried integrating by parts, but that didn't seem to do any good. Does anyone know a good way to start this problem?
 
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  • #3
Are we restricted to the reals? If so, then the limits of integration are not in the domain.
 
  • #4
Substituting x=t2 does not yield a Gaussian. As sennyk noted, the integration limits cannot be -infinity to infinity.
 
  • #5
break the integral into two parts one from -inf to 0 and then 0 to inf...if any of those two intregrals diverge then the whole thing diverges
 
  • #6
Midy1420 said:
break the integral into two parts one from -inf to 0 and then 0 to inf...if any of those two intregrals diverge then the whole thing diverges

The problem is more that the function isn't even defined on one of those intervals.
 

FAQ: Integrate x^(1/2)*e^(-x): Tips & Tricks

What is the best way to integrate x^(1/2)*e^(-x)?

The best way to integrate x^(1/2)*e^(-x) is by using integration by parts. You can let u = x^(1/2) and dv = e^(-x) which will result in du = (1/2)x^(-1/2)dx and v = -e^(-x).

How do I choose the limits of integration?

The limits of integration can be chosen based on the specific problem you are trying to solve. Typically, you would choose the limits based on the range of values for x that are relevant to the problem at hand.

Can I use substitution to integrate x^(1/2)*e^(-x)?

Yes, you can use substitution to integrate x^(1/2)*e^(-x). You can let u = -x which will result in du = -dx and x = -u. This can simplify the integration process.

Is there a shortcut method for integrating x^(1/2)*e^(-x)?

There is no specific shortcut method for integrating x^(1/2)*e^(-x). However, you can use properties of logarithms and exponents to simplify the integral and make it easier to solve.

Can I use a calculator to integrate x^(1/2)*e^(-x)?

Yes, you can use a graphing calculator or online integration calculator to find the definite integral of x^(1/2)*e^(-x). However, it is important to understand the steps involved in solving the integral by hand in order to fully grasp the concept and be able to apply it in other problems.

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