The main aim of this work can be classified into four objects. These are summarized as follows: The first objective, is to study some famous inequalities in a pre-Hilbert space, like Cauchy-Schwarz inequality, triangle inequality and Bessel's inequality with their reverses.The second objective, is to classify the one-dimensional integral inequalities and it's generalization to the multi-dimensional integral inequalities. The third objective, is to find explicit bounds for the unknown function in the Volterra integral inequalities. The fourth objective is to use the integral inequalities in studying the qualitative properties of solutions for the ordinary differential equations, fractional differential equations and fractional integro-differential equations.