This paper is devoted to studying the existence of of renormalized solution for an initial boundary problem of a quasilinear parabolic problem with variable exponent and $ L ^{1} $-data of the type \begin{equation*} \left\{ \begin{array}{ll} (b(u))_{t}-\text{div}(\left\vert \nabla u\right\vert ^{p(x)-2}\nabla u)+\lambda \left\vert u\right\vert ^{p(x)-2}u=f(x,t,u) \text{ } & \text{in}\hspace{0.5cm}Q=\Omega \times ]0,T[, \\ u=0 & \text{on}\hspace{0.5cm}\Sigma =\partial \Omega \times ]0,T[, \\ b(u)(t=0)=b(u_{0}) & \text{in}\hspace{0.5cm}\Omega , \\ & \end{array}% \right. \end{equation*}% where $ \lambda>0$ and $ T $ is positive constant. The results of the problem discussed can be applied to a variety of different fields in applied mathematics for example in elastic mechanics, image processing and electro-rheological fluid dynamics, etc.