Integrals are the values of the function found by the process of integration. It can be thought of as the process of getting f(x) from f'(x), the derivative.
Integration is absolutely fundamental for developing an understanding of calculus. It can be used to solve complex problems in mathematics and physics, especially those involving calculations of area in nontraditional shapes or curves, or to calculate the volumes of solids. In kinematics, integrals might be used to solve for displacement, velocity, and/or time. In probability, they may be be used to determine the chances of a variable landing within a certain range.