The Substitution Rule

The Substitution Rule is basically the reverse of the Chain Rule.

You find a sub-function within the function you are integrating, and represent it with a new variable. (The book, and Prof. Newberry too, uses u to represent the function).

Before you rewrite $g(f(x))dx  as  $u du , * make sure you go through and evaluate du:

du/dx = _    =>    du = _dx

If you're lucky, the value multiplied by dx when evaluating du will account for some of the things the function represented by u was multiplied by. Whether it does or doesn't, you will need to divide the remaining parts of the original function by the right side of the equation evaluating du to cancel out the multiplication by du.

To make this a bit clearer, I'll do Section 5.5 Example 2:

Find   Int = $ sqrt(2x+1) dx 

For u = 2x +1 ,

du/dx = 2   =>   du = 2 dx  => dx = du/2

Int = $ sqrt(u) du/2  =  1/2 $ u^(1/2) du

= (1/2) * u^(3/2) / (3/2)  + C

= (1/3) * u^(3/2)  + C

= (1/3) * (2x+1)^(3/2)  + C

If you took the trouble to read the above, you will have seen that in the first step of solving the integral using u, du was divided by 2 to correct for the fact that du/dx = 2 . 

I hope this helps.

*I don't have an integral symbol on my keyboard (who does?), so I'm using $ instead of the vertically stretched S.