I've seen occasions such as the xkcd comic
Or the classic - simplifying $16/64$ by cancelling out the six's to get $1/4$.
I was grading some intro-to-calc papers recently, and one student did something like the following:
$\int x\sqrt{1 + x^2} = \frac{x^2}{2} \sqrt{ 1 + \frac{x^3}{3} } + C$
How hard would it be to construct an integral of a function such that doing this 'naive' sort of integration (I suppose we are distributing the integral operator, in particular) will actually return the correct answer? I played around briefly, and it seemed like an interesting challenge.