I came across two approaches to the method of integration by substitution (in two different books).
Let $I=\int f(\phi(x))\phi'(x) dx$
$\therefore I=\int f(z)dz$
Let $I=\int f(x) dx$
$\therefore I=\int f(\phi(z))\phi'(z) dz$
My problem: While i can understand Approach I, I cannot understand Approach II. What is the difference between the two approaches. What is the difference in their applicability and usage? I am very confused. Please help.