We present studies on the low-temperature magnetization of a two-dimensional electron system (2DES) in the integer quantum Hall-effect regime. The 2DES was formed in a modulation-doped AlGaAs/GaAs heterojunction. Using molecular-beam epitaxy it has been integrated into a highly sensitive micromechanical cantilever magnetometer. We observe de Haas–van Alphen oscillations at even filling factors up to ν=40 which for ν<~20 are almost perfectly sawtoothlike. Oscillations at odd filling factors ν=3,5,7, and 9 are due to the spin splitting of Landau levels. For a quantitative analysis of our data, calculations of the magnetization based on a model density of states (DOS) have been performed. We describe the DOS by narrow Gaussian broadened Landau levels with an energy-independent background. In particular we find that this background DOS increases linearly with ν. This behavior is qualitatively explained using the concept of edge channels. From our data we evaluate the level broadening and, at odd filling factors, the exchange-energy contribution to the spin splitting of Landau levels.