This procedure is for **indefinite integrals**. See the
procedure for definite integrals for how
to handle integration with limits by substitution.

Procedure:

- 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. - Substitute
*u*for*g*(*x*) and*du*for( in the integral.*du*/*dx*)*dx*You should be left with

**no***x*'s in the integral. If you have*x*'s left over, you may be able to solve for*x*in your equation for*u*and replace them, but this doesn't always work.If you end up with extra

*x*'s, or without the proper*du*/*dx*, then you should go back and try a different*u*. - Compute the integral in terms of
*u*. - Replace
*u*by*g*(*x*) in the final result to get the answer in terms of*x*.

Note: there is no guaranteed way to choose the proper *u*; it takes
experience and sometimes a little creativity. Keep trying if it doesn't
work at first.

Note: the method of substitution does not work for all integrals (only on those functions that are the result of a chain rule derivative; not all functions are). If you can't get a substitution to work, try another integration technique.

Comments to: dpvc@union.edu

Created: Nov 3 1997 --- Last modified: Nov 11, 1998 10:51:00 AM