In this work the general theorems on families of functions depending on a parameter will be applied to the type of family most frequently encountered in analysis - integrals depending on a parameter. We present properties of the integrals depending on a parameter and discuss its possible applicability. The work is divided in three chapters. First we provide the definition and the concept of an integral depending on a parameter. In Chapter 1 we prove results concerning the continuity, differentiation and integration of such kind of integrals. Correspondingly, we give examples for each property mentioned in this Chapter. Chapter 2 deals with properties of improper integrals depending on a parameter. We provide the rules for passage to a limit under the sign and continuity of an improper integral depending on a parameter. Furthermore, we provide the results concerning the differentiation and integration of such types of integrals. The third chapter is concerned with special integrals depending on a parameter, called The Eulerian integrals, namely the Beta Function and the Gamma Function. We provide the analysis of these two functions in view of previously presented results. At the end of this work we give some examples of integral that can be calculated by employing this two functions.