Let $0\neq p(x)$ be a nondecreasing real valued differentiable function on $[0,\infty)$ such that $p(0)=0$ and $p(x)\to\infty$ as $x\to\infty$. Given a real valued function $f(x)$ which is continuous on $[0,\infty)$ and \[ s(x)=ıt_0^x f(t)\,dt. \] We define the weighted mean of $s(x)$ as \[ igma_p(x)=\frac1{p(x)}ıt_0^x p'(t)s(t)\,dt, \] where $p'(t)$ is derivative of $p(t)$. It is known that if the limit $\lim\limits_{x\to\infty}s(x)=s$ exists, then $\lim\limits_{x\to\infty}\sigma_p(x)=s$ also exists. However, the converse is not always true. Adding some suitable conditions to existence of $\lim\limits_{x\to\infty}\sigma_p(x)$ which are called Tauberian conditions may imply convergence of the integral $\int_0^\infty f(t)\,dt$. In this work, we give some classical type Tauberian theorems to retrieve convergence of $s(x)$ out of weighted mean integrability of $s(x)$ with some Tauberian conditions.