In this paper, a new test criterion for binary decision problems is proposed. The integrated power flunction over a parameter interval is first itroduced as an extension of the power function. The concept of the most integrated powerful (MIP) test based on the integrated power function is then introduced. The MIP criterion is to masimize the value of the integrated power function in any paricular parameter interval. As an applicationof the MIP test, the known signal detection problem is considered. The test statistic of the MIP detector for known signals is obtained and an approximation to the MIP test statistic is also considered.