Differentiation under the integral sign needs a theorem with a proof. There are several versions of the theorem. The best proofs I know of are from the dominated convergence theorem of measure theory.

For the Putnam exam, yes, it's challenging. It's so challenging that doing the work to do well on the exam is comparable with creating and publishing peer reviewed original research in math, and such a publication will likely do much more for the academic progress for a student than the Putnam.

Indeed, soon enough in math graduate school at a good research university, a student learns that everyone admits that no one can carry all the library around between their ears and doesn't try to. Instead, the three most important measures of progress are, in no particular order, research, research, and research.

Finally, IIRC, how to do integration of algebraic expressions in closed form has long been a solved problem and long ago was coded up in some software -- IIRC, the software will report the result or state that no closed form solution exists.