On a Class of Integrals related to Wallis Integrals

In this article we provide a motivation for studying integrals ${\displaystyle {\int }_0^{\frac{\pi }{2}}x{\cos }^{m}xdx}$, $m\in \mathbb{N}$. We demonstrate several properties of this class of integrals and show that it is closely related to the class of Wallis integrals, ${\displaystyle {\int }_0^{\frac{\pi }{2}}{\cos }^{m}xdx}$, $m\in \mathbb{N}$.