Two integrals I am trying to solve without closed form antiderivatives

[Ed Quanta]

How do I solve the integral of the functions x*exp(-a(x-b)^2) and x^2(exp(-a(x-b)^2) where a and b are positive real numbers?

I tried integration by parts and cannot think of how to find the integral. In addition, while I was able to find exp(-a(x-b)^2) in an integral table, the two functions I listed above, I was not able to find. Any hints, suggestions, or integration tables you can point me towards would be appreciated.

 

[arildno]

Set $$u=\sqrt{a}(x-b)\to{dx}=\frac{du}{\sqrt{a}}$$
Thus, for example, your first integrand transforms as:
$$\int{x}e^{-a(x-b)^{2}}dx=\frac{1}{a}\int(u+\sqrt{a}b)e^{-u^{2}}du=\frac{1}{a}\int{u}e^{-u^{2}}du+\frac{b}{\sqrt{a}}\int{e}^{-u^{2}}du$$

the first term is easily integrated, the other is seen to be the error function integral that is well tabulated.

 

[Ed Quanta]

Thanks.

I looked up the second term in an integration table (no problem). The first term I was able to integrate using substitution v = u^2 and dv=2udu. Here is my problem though. Can I solve this term as a definite integral with the bounds positive infinity and negative infinity? If so, how? I know the first term is going to equal -1/2a*exp(-v). But if I set the limit equal to negative infinity, this goes infinity. Does this term have a definite value?

 

[dextercioby]

Nothing will go to infinity if you do the calculations correctly.