A concept of fundamental integrals of one-dimensional harmonic oscillator is presented, and other integrals can be constructed by use of fundamental integrals. The above concept and method are extended to multidimensional harmonic oscillators. By directly constructing other integrals from the fundamental integrals, it is proved that there are three independent time-independent integrals for all kinds of two-dimensional harmonic oscillators and there are 2n-1 independent time-independent integrals for n-dimensional harmonic oscillators. The characteristics of the anisotropic two-dimensional harmonic oscillator is discussed when the ratio between two frequencies is rational or irrational number.