Abstract: This paper established a new existence and uniqueness result for solutions to one-dimensional backward stochastic differential equations (BSDEs) with only integrable parameters, where the generator $g$ is monotonic in $y$ and $\alpha$-H\{o}lder ($0\alpha1$) continuous in $z$. By Tanaka's formula and Girsanov's theorem we established a comparison theorem for solutions in $L^1$ to BSDEs, from which the uniqueness follows. By convolution technique we obtained a uniform approximation sequence of the generator $g$ and then constructed a sequence of solutions in $L^1$ for BSDEs. Finally, we proved the limitation of this sequence of solutions is the desired solution. This proved the existence.