Let \(B_{t}\) be a standard Brownian motion started at 0 . Use that for any function \(f\) we have:

\[\mathbf{E}\left[f\left(B_{t}\right)\right]=\frac{1}{\sqrt{2 \pi t}} \int_{-\infty}^{\infty} f(x) e^{-\frac{x^{2}}{2 t}} d x\]

to calculate:

\[\mathbf{E}\left[B_{t}^{2 k}\right]\]

for some \(k\), an integer. As a hint, you may want to use integration by parts and induction to come up with a formula for \(\mathbf{E}\left[B_{t}^{2 k}\right]\).