One venerable technique in the solution of differential equations is the `method of undetermined coefficients'. You guess the form of the answer, with some unknown coefficients, and then use the problem under discussion to discover the coefficients. For integration, the technique might go something like this:
is to be found. We guess that the answer is of the form
and differentiate both the integral and our guess to get the following:
and we set all the coefficients to zero, which gives us a (triangular) linear system for the unknown coefficients. This gives us
so our answer is
which is correct.
This can be done more efficiently for some problems. See the worksheet
undet.ms for the next section.