In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem.
The Laplace transform of a function $f(t)$, for $t \geq 0$ is defined by $\int_{0}^{\infty} f(t) e^{-st} dt$.
How to avoid looking at this definition as "magical"? How to somehow discover it from more basic definitions?
What is also very interesting is that the Laplace transform is nothing else but the continuous version of power series - see this insightful video lecture from MIT:
https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/resources/lecture-19-introduction-to-the-laplace-transform/
