I was thinking of proving $\int_{0}^{\infty} \sin(x) dx$ is unbounded?
Graphically the areas get added below the curve, but it seems to be adding equal positive and negative areas, just like $+A -A +A -A +A -A +....$ where $A$ represents the area between the curve and the x coordinates $0$ to $\pi$ but why the definite integral is unbounded?. Is it is bounded though?
Next I thought of proving it by contradiction, like let the definite integral be bounded then I was thinking how do we get a contradiction?
It's not unbounded. However, the limit $\displaystyle\lim_{x\to\infty} \int_0^x \sin t \,dt$ does not exist as you point out.