Partly Fun Integrals

Purpose

The purpose of this lab is to give you experience with using Maple to evaluate certain kinds of integrals.

Background

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.

  > with(student):

Warning: new definition for   D

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

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

Exercises

1. Use integration by parts, or otherwise, to perform the following integrations. Make your choices for u and dv clear.
   1. 
   2. 
   3. 
2. Suppose m and n are positive integers. Can you conjecture general formulas for the following integrals? Actually proving your conjectures would require the use of mathematical induction, and I'm not asking you to go that far. You may want to check out Maple's assume command.
   1. 
   2. 
   3. 
   4. 
3. Find an integral of simple form that Maple can't do. It shouldn't be hard, since most integrals can't be done analytically. (If Maple can't do an integral analytically, then it just returns the integral unevaluated.)

   A definite integral can be evaluated numerically, by using the evalf command outside the int command. Try this with your integral. For your report, come up with at least two reasons why an analytical answer is better than a numerical one, if you can get it.

4. In differential equations, one often has to solve for what is called the particular solution. For example a particular solution of the differential equation

   

   is . What we mean by solution is that the differential equation is satisfied if you substitute into it. Find a particular solution of the differential equation

   

   Hint - try , where A and B are constants. Can you choose values of A and B so that satisfies the differential equation?