In the days before computers, people used to put a tremendous amount of effort into developing techniques that would allow analytical evaluation of integrals. Still, no matter how they tried, many integrals refused to submit. With the advent of computers and numerical techniques for evaluating integrals, the importance of many of these techniques has diminished considerably.

However, there are some fairly simple techniques that are useful in solving practical problems. One of these, which also has many theoretical applications, is integration by parts. The standard formula is

The trick is to choose **u** and **dv** so that can be
evaluated. A standard example is an integral of the form

where **k** is a positive integer.
If is a function that is easy to integrate, and is an
anti-derivative of , then a single application of integration by parts
leads to

Note that the power of **x** has been reduced by one. If integration by
parts can be applied again, further reduction is possible. (What
determines whether we can apply integration by parts or not? Hint - it
is a property of .)

Maple has a convenient function for doing integration by parts in the
`student` package. The command is `intparts` and examples are
shown below. Remember that you have to load the `student` package
before you can use the `Int` and `intparts` commands.

The `intparts` command takes two arguments. The first is one is
the integral, which must be expressed using the `Int` command, and
the second is the the function . Note also that the `
intparts` command doesn't do the final integral, but you should be able
to do that one in your head.

> with(student):

Warning: new definition for D

> intparts(Int(x*cos(x),x),cos(x));

> intparts(Int(x*cos(x),x),x);

It is often necessary to integrate by parts more than once. This is
most easily accomplished by using nested `intparts` commands, as
shown below.

> intparts(Int(x^2*sin(x),x),x^2);

> intparts(intparts(Int(x^2*sin(x),x),x^2),x);

In this example two integrations by parts were required. They were
shown separately in the example, but only for the sake of
clarity. Probably the best strategy is to start with the innermost
integration by parts and then simply edit the command, adding further
`intparts` commands as needed.

Fri Aug 25 18:22:38 EDT 1995