In this work, Fourier integral microscope (FIMic), an ultimate design of 3Dintegral microscopy, is presented.By placing a multiplexing microlens array at the aperture stop of the microscope objective of the host microscope, FIMic shows extended depth of field and enhanced lateral resolution in comparison with regular integral microscopy.As FIMic directly produces a set of orthographic views of the 3D-micrometer-sized sample, it is suitable for real-time imaging.Following regular integral-imaging reconstruction algorithms, a 2.75-fold enhanced depth of field and 2 -time better spatial resolution in comparison with conventional integral microscopy is reported.Our claims are supported by theoretical analysis and experimental images of a resolution test target, cotton fibers, and in-vivo 3D-imaging of biological specimens.