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

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The integral \(\int_{-\infty}^{\infty} x^{1/2} e^{-x} \, dx\) cannot be evaluated directly due to the function's domain restrictions. The discussion highlights the necessity of breaking the integral into two parts: from \(-\infty\) to \(0\) and from \(0\) to \(\infty\). It is established that if either of these integrals diverges, the overall integral diverges as well. The substitution \(x=t^2\) is suggested but noted as ineffective for yielding a Gaussian form.

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I can't seem to figure this integral out
\intx^(1/2)*e^(-x) dx
from -\infty to \infty

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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Are we restricted to the reals? If so, then the limits of integration are not in the domain.
 
Substituting x=t2 does not yield a Gaussian. As sennyk noted, the integration limits cannot be -infinity to infinity.
 
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
 
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.
 

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