Integral Dilemma

I have provided a simple solution using graph.

First note that when we're integrating a function $y = f(x)$ we're finding out the total algebraic sum of area that it makes with the x-axis. In our case,

$\begin{cases} y = x \\ y = -x \end{cases}$

The area will be $\infty - \infty$ as shown below:

Calling $\infty - \infty\ = 0$ is meaningless as $\infty$ is not a number and you cannot perform arithmetic on it even though the areas shown above seem to be perfectly symmetrical to each other. The only valid explanation in this case would be that this area does not converge to a particular sensible value.