On a Class of Integrals related to Wallis Integrals

In this article we provide a motivation for studying integrals $\displaystyle \int_0^{\frac{\pi}{2}}x\cos^mx~dx$, $m\in \mathbb{N}$. We demonstrate several properties of this class of integrals and show that it is closely related to the class of Wallis integrals, $\displaystyle \int_0^{\frac{\pi}{2}}\cos^mx~dx$, $m\in \mathbb{N}$.