if i do $\displaystyle \int \frac{dy}{dx} dy $ what does this go to... y?
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No. If it was $\displaystyle \int \frac{dy}{dx}~dx$ then it would be $\displaystyle y + C$. If y is a function of x, then $\displaystyle \frac{dy}{dx}$ is a function of x too. So $\displaystyle \int \frac{dy}{dx}~dy = \frac{dy}{dx}\cdot y$
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