Find .

First, we note that since *a* only appears as an *a*^{2} term, *I*(*a*) is an even function (), and so we can confine our work to .

Next, we may proceed by two different methods previously used here. The first method is differentiation under the integral sign. Taking the derivative with respect to *a*, we see

.

Now, making u-substitution , *a*>0, we see , and , and so our above integral becomes

.

This first order differential equation has general solution .

Now, for *a*=0, we have the integral of a Gaussian

,

and thus , and .

For the second method, we first note that the integrand is an even function of *x*, and so

.

Next, one can see that

,

and thus

,

and so

.

I previously proved that for *a*>0,

,

and therefore

,

for *a*>0, and so, for all real *a*, we have

,

as with our other method.

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Tags: Calculus, Gaussian Integral, integrals, Math, Monday Math

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