sequences and series - Infinite sum of positive integers and why an integral test doesn't disprove it

Problem

I first came across this statement

$\sum_{n=1}^\infty n = -\frac{1}{12}$ a couple of years ago.

Why does an integral test for convergance not disprove this.
That is, with an integral of
$$\int_0^\infty x dx$$

I see the integral test requires a monotonically decreasing function and this will be why you can't use it for the infinte sum of positive integers.
I fail to understand why this function requires a decreasing function. Perhaps this is beyond the scope of what I'm trying to understand.

Context

I saw a video challenging the viewer to solve the Balancing bricks problem and I proceeded to solve it to the point of the Sum of the harmonic series which I looked up information.

I saw the integral test as a way to show this series also had an infinite sum as demonstrated in the Harmonic Series Wikipedia Article. This immediately reminded me about that ol' friend $\sum_{n=1}^\infty n = -\frac{1}{12}$ and I was lead to wonder why that integral test couldn't be applied. Cue much internet searching and now this question.