Change of variable in integration

Question

When integrating the following $$y = \int \frac{dy}{dx}\,dx$$ We can apply change of variable which is based on the chain rule by multiplying $\frac{dx}{du}$ on the RHS. $$y = \int \frac{dy}{dx}·\frac{dx}{du}\,du$$ But why does the equation still holds? I think the left hand side should also be affected. Because $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$ but $\frac{dy}{dx} \not= \frac{dy}{dx}\frac{dx}{du}$. I want to solve it on my own but in order to understand rigorous process it seems I should learn more advanced concepts like differential form. And as I understand there are at least two standpoints about infinitesimal and limit for interpreting $\int \frac{dy}{dx}\,dx$. How to understand this?

Answer 1

Because $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$ but $\frac{dy}{dx} \not= \frac{dy}{dx}\frac{dx}{du}$.

Indeed, but $\frac{dy}{du} = \frac{dy}{dx}\frac{dx}{du}$ .$$\begin{align}\int\mathrm d y&=\int\dfrac{\mathrm d y}{\mathrm d u}\mathrm d u\\&=\int\dfrac{\mathrm dy}{\mathrm d x}\dfrac{\mathrm d x}{\mathrm d u}\mathrm d u\end{align}$$