Integrating both sides of equation

Question

I'm given that -> $$\frac{dy(x)}{dx}=x$$ I integrate both sides -> $$\int \frac{dy(x)}{dx}dx=\int x\,dx$$

Would it be correct if i canceled out the $dx$s and wrote -> $$\int dy=\int x\, dx$$

therefore

$$y= \frac{x^2}{2}+C,$$ where $C\in \mathbb{R}$

Answer 1

Yes, this would be fine. It's just like integration by substitution. $$\int y'(x) \,dx$$ Substitute $u=y(x)$. Then $du=y'(x)\,dx$, which is exactly equal to the integrand. So you simply get $$\int \,du\equiv\int\,dy$$