TITLE: Different ways to evaluate $\int_0^\infty xI_0(2x)e^{-x^2}\,dx$
QUESTION [7 upvotes]: Evaluate $$\int_0^\infty xI_0(2x)e^{-x^2}\,dx$$ where $$I_0(x) = \frac 1\pi \int_0^\pi e^{x\cos\theta}\,d\theta$$ is a Bessel Function. 
Source: Berkeley Math Tournament 
This question was on a math contest for high school students, so I am looking for other methods that preferably do not involve higher mathematics than Calc II. However, I am also interested in other ways to solve this problem that goes beyond the normal calculus curriculum. My solution is posted below as an answer.

REPLY [3 votes]: I thought it might be instructive to present a way forward the relies on only the series expansion of the exponential function, evaluating two integrals using reduction formulae, and straightforward arithmetic.  To that end, we proceed.

Using the Taylor series for $e^t=\sum_{n=0}^\infty \frac{t^n}{n!}$, with $t=2x\cos(\theta)$, we can write $I_0(2x)$ as
$$\begin{align}
I_0(2x)&=\frac1\pi \int_0^\pi e^{2x\cos(\theta)}\,d\theta\\\\
&=\frac1\pi \sum_{n=0}^\infty \frac{(2x)^n}{n!}\int_0^\pi \cos^n(\theta)\,d\theta\tag 1
\end{align}$$
Next, using the reduction formula for $\int_0^\pi \cos^n(\theta)\,d\theta=\frac{n-1}{n}\int_0^\pi \cos^{n-2}(\theta)\,d\theta$ reveals
$$\int_0^\pi \cos^n(\theta)\,d\theta=\begin{cases}\pi\frac{n!}{(n!!)^2}&,n\,\text{even}\\\\0&,n\,\text{odd}\tag2\end{cases}$$
Using $(1)$ and $(2)$, we find that 
$$\begin{align}
\int_0^\infty xe^{-x^2}I_0(2x)\,dx&= \frac1\pi\sum_{n=0}^\infty \underbrace{\frac{4^n}{(2n)!}\left(\pi\,\frac{(2n)!}{((2n)!!)^2}\right)}_{=\frac{\pi}{(n!)^2}}\,\,\underbrace{\int_0^\infty x^{2n+1}e^{-x^2}\,dx}_{=\frac12 n!}\\\\
&=\frac12\sum_{n=0}^\infty \frac{1}{n!}\\\\
&=\frac{e}{2}
\end{align}$$
where we used the reduction formula $\int_0^\infty x^{2n+1}e^{-x^2}\,dx=n\int_0^\infty x^{2n-1}e^{-x^2}\,dx$, along with the elementary integral $\int_0^\infty xe^{-x^2}\,dx=\frac12$, to establish $\int_0^\infty x^{2n+1}e^{-x^2}\,dx=\frac12 n!$. 

Tools Used:  The Taylor Series for $e^x$, the reduction formula for $\int_0^\pi \cos^n(x)\,dx$, and the reduction formula for $\int_0^\infty x^{2n+1}e^{-x^2}\,dx$