Integral of $dy/dx$ confusion Why is $\displaystyle \int \dfrac{dy}{dx} dx = y + c$, but for example $\displaystyle \int \dfrac{dy}{dx} y dx = \dfrac{1}{2} y^2 + c$ instead of $\dfrac{1}{2} y^3 + c$?
Using and abusing the mathematical notation as sometimes is done when dealing with differential equations, what you really have here is

$$\int\frac{dy}{\color{red}{dx}}\color{red}{dx}=\int 1\cdot dy=y+C\;,\;\;\text{since}\;\;\frac{d}{dy}(y+C)=1$$

OTOH, 

$$\int\frac{dy}{\color{red}{dx}}y\,\color{red}{dx}=\int y\,dy=\frac{y^2}2+C\;,\;\;\text{since}\;\frac d{dy}\left(\frac12y^2+C\right)=y$$
I have another question and i hope someone can clarify it:
How do you find out what this is: $\int \frac{dy}{dx}$