Do the following integrals:

1. \[ \int_0^1 x^3 dx\]
2. (*) Hint: integrate by parts. \[\int_0^\infty x \exp(-x) dx\]
3. (**) \[\int_0^\infty  \exp(-\frac12 x^2) dx\] Try looking at  \[\Big(\int_0^\infty  \exp(-\frac12 x^2) dx\Big)\Big(\int_0^\infty  \exp(-\frac12 y^2) dy\Big)=\int_0^\infty \int_0^\infty  \exp(-\frac12 x^2) \exp(-\frac12 y^2)dx dy\] and then changing to polar coordinates. How des this help you figure out the original question ?