Often while solving an equation of the form

$$ f(y) \frac{dy}{dx} = g(x) $$

we integrate on both sides to obtain

$$

\int f(y) \frac{dy}{dx} \, dx = \int g(x) \, dx.

$$

We normally write it as

$$

\int f(y) \, dy = \int g(x) \, dx.

$$

But I don't understand how we replace $ \int f(y) \frac{dy}{dx} \, dx $ with $ \int f(y) \, dy $ on the LHS.

Is there a way to show that

$$

\int f(y) \frac{dy}{dx} \, dx = \int f(y) \, dy

$$

## Best Answer

This is just integration by substitution. Note for $$ \int f(y)y' dx $$ we can set ${u = y(x)}$ to obtain $$ \rightarrow \int f(u)du $$ but ${u=y}$, so $$ \rightarrow \int f(y)dy $$