Lab 2: Patterns in Integrals
Seattle Pacific University, MAT 1235, Calculus II
Objectives


To use Maple to explore the antiderivatives of some families of functions and to
recognize patterns in the forms of these antiderivatives.
To get practice with drawing conclusions from patterns of answers.
Due Date: Monday 2:00 p.m.

Do not wait until the last minute to finish the lab. You never know what technical
problems you may encounter (no papers in the printer, electricity is out, your best
friend call and talk for 4 hours, alien attack…etc).
Name 1:
Name 2:
Date:
The original version of this lab was adapted by Brian Gill and the SPU mathematics department from Lab 17
(Patterns in Integrals, by Bonnie Gold and Anita Solow) in Learning by Discovery: A Lab Manual for Calculus,
Anita Solow, editor, 1993. The book is volume 27 of the MAA Notes series published by the Mathematical
Association of America.
Maple Review: Indefinite Integrals
1. Suppose we want to evaluate the integral
1
 x dx .
The Maple comment is:
>int(1/x,x);
ln(x)
Note that the integration constant C as well as the absolute value within the natural log function
are not shown in the Maple result, you need to put it back when typing your answers.
2. The exponential function e x is represented in Maple as
> exp(x);
e
x
2
Lab Exercises
In this lab you will use a computer to calculate antiderivatives of several functions. Each
exercise concerns a family of related functions. You are to develop an intuition about the form
of the antiderivatives of these families.
In order to explain some of your answers, you may need to do additional Maple experiments.
Make sure you state the results clearly in your report.
1. a. Use Maple to evaluate the following integrals of rational functions.
Indefinite Integrals
Answers
1
  x  1 x  2dx
1
  x  2 x  7 dx
1
  x  2 x  7 dx
1
 x  x  3dx
1
  x    x   dx
1
2
1
  x  3
2
dx
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b. (i) Based on the pattern of your answers in part (a), make a conjecture about the value of
1
the indefinite integral 
dx where a and b are constants and a  b .
 x  a  x  b 
∫
1
𝑑𝑥 =
(𝑥 +∫ 𝑎)(𝑥 + 𝑏)
(ii) Does your formula work when a or b is negative? Explain/provide evidence.
(iii) Do a and b need to be integers for the formula to work? Explain/provide evidence.
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(iv) Does your formula work when a = b? Explain/provide evidence.
If there are any exceptions, can you guess a separate formula that will work in those cases?
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2. a. Use Maple to evaluate the following integrals of powers of x times the natural logarithm.
Indefinite Integrals
Answers
 ln xdx
 x ln xdx
x
x
x
2
ln xdx
3
ln xdx
5
ln xdx
b. (i) Based on the pattern of your answers in part (a), make a conjecture about the value of
the integral  x n ln xdx where n is a constant.
∫ 𝑥 𝑛 𝑙𝑛𝑥 𝑑𝑥 =
(ii) Is the formula valid if n = 0? Explain/provide evidence.
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(iii) Does it work for all negative values of n? Explain/provide evidence.
(iv) Does it work if n is not an integer? Explain/provide evidence.
If there are any exceptions, can you guess a separate formula that will work in those cases?
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3. a. Use Maple to evaluate the following integrals of powers of x times the exponential function.
Indefinite Integrals
Answers
 xe dx
x
 x e dx
2 x
 x e dx
3 x
 x e dx
4 x
b. Based on you answers above, make a conjecture about the form of the antiderivative of
x n e x when n is a positive integer. When giving your answer, do not worry about the exact
values of the constants involved, just give the general form of the functions.
For example, if you think the integral of x n f ( x) is a degree n polynomial, you write
x
where a0 , a1 ,
n
f ( x)dx  a0  a1 x  a2 x 2 
 an x n
, an are constants.
∫ 𝑥 𝑛 𝑒 𝑥 𝑑𝑥 =
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