  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, $\displaystyle
\int_0^{\frac{\pi}{2}}\cos^mx~dx$, $m\in \mathbb{N}$.
