## Monday Math 122

Find .

First, we note that since a only appears as an a2 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.