The definite integral of the multiplication of two functions corresponds to the inner product of the two (multiply pointwise, accumulate the result. The indefinite integral traces the value of the definite integral as the limit increases, and allows understanding its behavior.
Orthogonal functions are useful as basis functions for representing signals in different bases than the canonical (cartesian) basis. The best know example is the Fourier series of orthogonal functions, sines and cosines of increasing frequencies.
