Integration by Substitution

Integration by Substitution is "sort of the reverse Chain Rule".

To integrate using substitution, step one is to choose "u." "u" is a piece of the equation that you want to substitute. Most often, if there is a "ln(x)," choose that to substitute. For example in the equation below, "u" is the part in the parenthesis.

Step 2: Now that we have our "u," we want to take the derivative of it.

u= x²-1
du = 2x*dx

Now that we have du, we want to isolate dx so we can plug it back into the original.

du = 2x*dx
2x = 2x
dx= du
         2x

Step 3: Plug dx into the original function and replace the part of the equation that you used for "u" with "u." 

If you look closely, it appears that the 2x's cancel out and you are left with:

                                                              ∫u4du

Step 4: Now you know how to take the integral of this kind of function and you get:

Step 5: This may look complete, but the "u" is just something we substituted for, so we 

want to replace "u" with what we previously defined it as. 

Completed, the integral would be: