By introducing fundamental integrals of one-dimensional linear damped oscillators the other first integrals can be constructed, including time-irrelevant integrals. The above method is extended to multidimensional systems, in order to construct different integrals of two-dimensional and n-dimensional linear damped oscillators. It is proved that there are three independent time-irrelevant integrals for all kinds of two-dimensional linear damped oscillators, and 2n-1 independent time-irrelevant integrals for n-dimensional linear damped oscillators. Using the transformation of variables the first integrals of linear damped oscillator transform into ones of harmonic oscillator.