I am reading the Separation of variables page on wikipedia, at a certain point it states that the following equation
Is equal to (1) because of the substitution rule of integrals.
The substitution rule of integrals linked is the following:
I am failing to see why the first two equations are the same given this rule, it seems to me that the left hand side of (1) ,following the notation of the substitution rule, can be seen as:
$$\int{\frac{1} {f(\phi(t))} \phi'(t) }dt$$ and the left hand side of the first equation from the top can be seen as:
$$\int\frac{1} {f(\phi(t))}d \phi$$
So I can't see how we apply the substitution rule.
Mind to help me out on why the first two equations are the same with the substitution rule?
Let $f(y) = \frac{1}{h(y)}$ for whatever $h$ you're interested in. Then the substitution rule, as stated by wikipedia (although without the limits) say that $$ \int\frac{1}{h(y)} dy = \int f(y)\, dy = \int f(y(x))\cdot y'(x)\,dx = \int\frac{1}{h(y)}\frac{dy}{dx} dx $$ The first equality is just inserting the definition of $f$.
The second is applying the substitution rule exactly as quoted from wiki, except I swapped out $x \to y$ in the integral on the left-hand side, and $\phi \to y$ and $t\to x$ on the right-hand side.
The third equality I get by reinserting $1/h$ for $f$, change how I write the derivative of $y$ and make $y$'s dependence on $x$ implicit (it's still there, I just didn't write it).