Certain results in calculus are widely quoted but little proved.
Notable are:
- transcendence of $e$
- transcendence of $\pi$
- non-integrability of $e^{-x^2}$ (and others)
These are all impossibility / field theory,
and are little-proved (and the proofs are little-known
to practicing mathematicians) because the methods
are a little involved and lead nowhere (that we know):
they are long dead ends.
Interesting digressions, perhaps,
esp. Liouville's theorem
Lindemannâ€“Weierstrass theorem: $\pi$, and also $e$
(but more elementary proofs of $e$).