Two approaches to the method of integration by substitution I came across two approaches to the method of integration by substitution (in two different books).
Approach I
Let $I=\int f(\phi(x))\phi'(x) dx$
Let $z=\phi(x)$
$\therefore \phi'(x)dx=dz$
$\therefore I=\int f(z)dz$
Approach II
Let $I=\int f(x) dx$
Let $x=\phi(z)$
$dx=\phi'(z) dz$
$\therefore I=\int f(\phi(z))\phi'(z) dz$
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.

The two approaches are the same, but one taken forward and the other backward.
The first form is used when the factor $\phi'(x)$ seems obvious.
For instance, in
$$\int \sin x\cos x\,dx$$ you can use $\cos x=\sin'x$ and the integral becomes
$$\int z\,dz.$$
The second form is used when you hope that $f(\phi(z))$ will be simpler than $f(x)$.
For instance, you want to get rid of the square root in
$$\int \frac{\sqrt x}{x+1}dx$$
with the subsitution $x=\phi(z)=z^2$, giving
$$\int\frac{z}{z^2+1}2zdz=2\int\left(1-\frac1{z^2+1}\right)dz.$$
