<p>Integrals have a variety of applications in many different areas like mathematics, science, engineering, statistics and economics. Keeping this fact in view, in this investigation, we evaluate some interesting general unified infinite integrals associated with incomplete Aleph-functions by using a known integral, which are expressed in terms of incomplete Aleph-functions. Next, we give some corollaries in terms of other incomplete special functions by assigning some particular values to the parameters of incomplete Aleph-functions (for example, incomplete <i>I</i>-function, incomplete <i>H</i>-function). The results presented in this paper are unified and general in nature.</p>