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

- 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(2*x*+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*. - 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. - Compute the value of
*u*at the limits of integration.That is, compute

*u*when and when*x*=*a* . These will become the new limits of integration.*x*=*b* - Substitute
*u*for*g*(*x*) and*du*for( in the integral, and replace the limits of integration by the new one computed in the previous step.*du*/*dx*)*dx*You should be left with

**no***x*'s in the integral, and the limits of integration will have changed. - 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.

Comments to: dpvc@union.edu

Created: Nov 3 1997 --- Last modified: Nov 3, 1997 3:00:48 PM