  For real numbers $p,q>1$ we consider the following family of integrals:
\begin{equation*}
\int_{0}^{1}\frac{(x^{q-2}+1)\log\left(x^{mq}+1\right)}{x^q+1}{\rm d}x \quad
\mbox{and}\quad
\int_{0}^{1}\frac{(x^{pt-2}+1)\log\left(x^t+1\right)}{x^{pt}+1}{\rm d}x.
\end{equation*} We evaluate these integrals for all $m\in\mathbb{N}$, $q=2,3,4$
and $p=2,3$ explicitly. They recover some previously known integrals. We also
compute many integrals over the infinite interval $[0,\infty)$. Applying these
results we offer many new Euler- BBP- type sums.
