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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.

next up previous
Next: Exercises Up: Partly Fun Integrals Previous: Purpose

William W. Farr
Fri Aug 25 18:22:38 EDT 1995