The aim of this work is to design, analyze and test the behaviour of first-order CLIK (Closed-Loop Inverse Kinematics) manipulator algorithms under the influence of Cartesian integral error feedback. Although CLIK algorithms has been widely and successfully applied in robot manipulators with structured workspaces, in aerial manipulators the lack of structured workspaces is highly demanding for the control algorithm and some extra requirements are needed. Thus, in this work the standard proportional action in first-order CLIK algorithms is enhanced adding integral actions. Among others benefits, it provides a smoother behaviour needed for smart manipulation, zero steady-state tracking error, rejection to constant disturbances (like those of the numerical algorithm errors) and zero sensitivity to dc-gain uncertainties in stable systems. The theoretical achievement is corroborated with both simulations and experiments on a 7-DoF lightweight aerial manipulator. Experiments also include the integration with a SNS (Saturation in the Null Space) algorithm to deal with physical constraints, demonstrating a successful implementation and performance on a RTOS (Real-Time Operating System).