Evaluate the integral from standard from

[osker246]

Homework Statement

evaluate \inte^(\frac{\kappa*x^2}{2KT})dx with limits of integration from -infinity to +infinity using the standard form \inte^(-C*x²)dx = (\frac{\pi}{4C})^1/2 with limits of integration from 0 to +infinity. Note κ, k, and T are constants. In the standard form c indicates a constant. Note the function being integrated is an even function: f(x)=f(-x).

The Attempt at a Solution

Well looking at the equation I see C=\frac{-\kappa}{2KT}. I then plug C into (\frac{\pi}{4C})^1/2giving (\frac{-2KT\pi}{4\kappa})^1/2.

My next step would be to evaluate:
2*[(\frac{-2KT\pi}{4\kappa})^1/2]^{+infinity}_{0}

But I no longer have my variable x to do so, am I missing something? Is my answer simply (\frac{-2KT\pi}{4\kappa})^1/2?

 

[Unit]

Hint: exp(-Cx²) is an even function, symmetric about the y-axis.

 

[osker246]

Thanks for the reply. I did see that it was an even function, that's why I changed my limits of intergration to 0 to +infinity and multiply the answer by two. But the integral of the standard function doesn't keep X as a variable. So how do I evaluate this integral with my limits of intergration? I feel like I am over looking something here.

 

[dextercioby]

You needn't evaluate any limit. The integral you're given already takes into account the limits at infinity and 0.

 

[Unit]

Ah, my mistake osker246. And yes, dextercioby is right.

 

[osker246]

So does this mean my answer is 2*(\frac{-2KT\pi}{4\kappa})^1/2?