Matlab cannot evaluate gaussian integral with quadratic denominator

[ILens]

Hello!

I have the following problem: I cannot solve the integral below by the means of Matlab.

$$\int_{-\infty}^{\infty} \frac{e^{-t^2}}{\left(2-t\right)^2 + 16} dt$$​

When I write the following in Matlab

>> syms t;
>> y = exp(-t^2) / (16 + (2 - t)^2);
>> int(y, t, -inf, inf)

it gives me the output

Warning: Explicit integral could not be found.
> In sym.int at 58
 
ans =
 
int(exp(-t^2)/(16+(2-t)^2),t = -Inf .. Inf)

I managed to calculate the integral by the means of both Mathcad and Mathematica. Mathcad gave 0.088 as an answer (I had to explicitly specify "Infinite Limit" as a method). Mathematica gave me 0.0880741, I used the NIntegrate function:

$$\mbox{NIntegrate}\left[\frac{e^{-t^2}}{\left(2-t\right)^2 + 16}, \left\{t, -\infty, \infty \right\} \right]$$​

Does anyone have an idea, how I can solve this integral in Matlab? What do Mathcad and Mathematica use in order to solve it?

Thanks!

 

[dextercioby]

I think it/they use(s) the theorem of residues.The integrand has simple poles at $2\mp 4i$.

Daniel.

 

[dextercioby]

I didn't find this integral in G & R 5-th edition,CD version

$$\int_{0}^{\infty} \frac{e^{-ax^{2}+bx+c}}{x^{2}+d^{2}} \ dx$$

,but this one was

$$\int_{0}^{\infty} \frac{e^{-\mu^{2}x^{2}}}{x^{2}+b^{2}} \ dx$$

Daniel.

 

[ILens]

G & R 5-th edition,CD version

Could you please explain what "G & R" is?

 

[dextercioby]

Gradshtyn & Rytzhik,"Tables of Series,Integrals and Products",Academic Press,5-th edition,CD version.

Daniel.

 

[dextercioby]

Here it is,courtesy of Mathematica,a closely related integral.

$$\int_{-\infty}^{+\infty} \frac{e^{-x^{2}}}{(2-x)^{2}+4} \ dx =\frac{\sqrt{\pi}}{6}\left[3\sqrt{\pi}\cos 8-12 \ _{1}F_{2}\left(1,\frac{3}{4},\frac{5}{4};-16\right) +64 \ _{1}F_{2}\left(1,\frac{5}{4},\frac{7}{4};-16\right)\right]$$

Daniel.