During the course of is wordy for during. It’s slightly different from in the course of, which is often replaceable with other prepositions like in, over, and while. And while during is usually the best replacement for during the course of, in and on are sometimes better.
In each of these sentences, during the course of could be shortened to just during:
In other words, during the course of integration, the two companies will need to combine all of their operating processes.[Forbes]
During the course of yesterday’s proceedings it emerged that civil court proceedings are pending in the matter. [Irish Times]
Schubert interviewed nine players who were members of the 2010 squad during the course of his investigation. [Sydney Morning Herald]
During the course of the day of his arrest, agents surreptitiously brought in drug dogs who signaled traces of drugs. [The Province]
As these examples demonstrate, during the course of often appears in business writing, and it’s also common in bureaucratese. In most normal contexts, it sounds artificially formal.