Up: Notes for Math 13

# Notes on Definite Integrals by Substitution:

There are two methods for handling definite integrals using substitution. The first uses the Fundamental Theorem of Calculus: you first compute the indefinite integral (using the subtitution method for indefinite integrals), and then use the result as the antiderivative in the FTC (i.e., evaluate it at the limits of integration and subtract).

The second uses substitution to convert the entire integral into one in terms of a new variable, u, as follows:

1. Make a choice for u.
There is no set rule for how to make this choice, so you may have to make several tries. A good rule of thumb is that if there is a function of something other than x, (e.g., cos(2x+1)) or a power of some function of x or a root of a function of x, then these functions are good candidates for u.

2. Compute du/dx.
This expression should appear as part of the integral. If it doesn't, you may have chosen a bad u. If the integral is missing some constant multiplier, you can fix it by multiplying by this factor inside the integral and dividing by this factor outside the integral.

3. Compute the value of u at the limits of integration.
That is, compute u when x = a and when x = b. These will become the new limits of integration.

4. Substitute u for g(x) and du for (du/dx) dx in the integral, and replace the limits of integration by the new one computed in the previous step.
You should be left with no x's in the integral, and the limits of integration will have changed.

5. Compute the definite integral in terms of u as normal.
You never go back to x's using this method. Once you have converted the limits, there is no need to go back to the x's.

Note: The second method is the one I recommend, as it is usually shorter and easier to handle, provided you remember to adjust the limits of integration. If you don't, however, you will run into serious trouble! If this happens to you a lot, you may prefer the first method.

Up: Notes for Math 13