INTEGRATION BY PARTS

An integral of the form can usually done by the technique of integration by parts. The formula for this is given by

=

In using this technique we need to keep three rules of thumb in mind when selecting u and dv

1. u times dv should include everything inside the original integral.

2. It should be possible to find v the antiderivative of dv.

3. The integral of vdu should be "easier or no more difficult than the original problem".

In this worksheet we'll use MAPLE to go through integration by parts in as close to a step-by-step manner as we can. The package math191 contains a routine called intByParts. To make it available we must enter

> with(math191):

On the next line enter the expression f that we wish to integrate.

> f:=x*exp(x);

Now enter your choice for u.

> u:=x;

The command intByParts returns a list of steps used in the process. To be able to access these steps one at a time, we will give the output a name (such as s) and put a colon at the end of the command.

> s:=intByParts(f,u):

In the next step we'll be able to see the first step which is the selection made for u and dv

> s[1];

To see the second step enter

> s[2];

To get the third step

> s[3];

In this last step Maple will usually complete the problem (even if it would take us several steps to do it)

> s[4];

>

EXERCISES

Find antiderivatives of each of the following by using paper and pencil and then check your work at each step by using the routine outlined above.

To use the routine just move back up to the line where f is entered. Enter the new expression for f and then go on to enter a new expression for u. Execute the other lines to see the steps.

x*(exp(-2*x))

sqrt(x)*ln(x)

sin(x)*ln(cos(x))

and

x^3/sqrt(x^2+1)

>

Dan Symancyk June 1994 January 1997 February 1997 January 2000