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MATH 101

Summer 2013
Lecture Notes
Integrating by Parts
page 1
Sample Problems
Compute each of the following integrals. Please note that arcsin x is the same as sin
the same as tan 1 x
1.
Z
2.
Z
3.
Z
4.
Z
x
xe dx
x cos x dx
4x
xe
dx
ln x dx
5.
Z
arcsin x dx
6.
Z
arctan x dx
7
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
MATH 101

Summer 2013
Inferring a Gaussian distribution
Thomas P. Minka
1998 (revised 2001)
Abstract
A common question in statistical modeling is which out of a continuum of models are likely
to have generated this data? For the Gaussian class of models, this question can be a
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
MATH 101

Summer 2013
An example of integration by parts.
We want to nd the antiderivative of
2
x5 ex dx.
There is more than one way to do this. The point of this handout is to try two different
ways, and to practice combining substitution with integration by parts.
The rst me
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
MATH 101

Summer 2013
Tabular Method for Integration by parts
Example 1 Evaluate
x2 cos x dx
D
I
x2
2x
2
0
cos x
XXX +
XXX
z
X
XX
XXX
XX
z
XXX +
XXX
z
X
sin x
cos x
sin x
x2 cos x dx = x2 sin x 2x( cos x) + 2( sin x) + C
= x2 sin x + 2x cos x 2 sin x + C
Example 2 Evaluate
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
MATH 101

Summer 2013
Integration
by parts
A special rule, integration by parts, is available for integrating products of two functions.
This unit derives and illustrates this rule with a number of examples.
In order to master the techniques explained here it is vital that you
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
MATH 101

Summer 2013
Integration by parts
14.4
Introduction
Integration by parts is a technique for integrating products of functions. In this Block you
will learn to recognise when it is appropriate to use the technique and have the opportunity to
practise using it for ndin
College of the North Atlantic  Happy ValleyGoose Bay Campus
Stats
MATH 101

Fall 2016
Social Statistics
SOC 3112090
Problem set 3 (Measures of Variability): Two points in total.
Due by 11:59 pm on Sunday, June 5, 2016
Please answer ALL questions. Your score will be deducted 15% of one point for each subquestion unanswered.
1. An IQV of .3
College of the North Atlantic  Happy ValleyGoose Bay Campus
Stats
MATH 101

Fall 2016
MATH 1829 ELEMENTS OF STATISTICS
EXAM 1 REVIEW
Name_
Multiple Choice:
Identify the following as qualitative or quantitative:
1) Blood Pressure of a patient
A. Qualitative
B. Quantitative
Identify the following as an observational study or experimental des
College of the North Atlantic  Happy ValleyGoose Bay Campus
Stats
MATH 101

Fall 2016
Social Statistics
SOC 3112090
Problem set 4 (Normal Distribution): Two points in total.
Due by 11:59 pm on Sunday, Jun 12, 2016
Note: You must show your work and answer all questions. Or your score will be deducted
accordingly. Also, keep in mind that dr
College of the North Atlantic  Happy ValleyGoose Bay Campus
Stats
MATH 101

Fall 2016
Social Statistics
SOC 3112090
Problem set 2 (Measures of Central Tendency): Two points in total.
Due by 11:59 pm on Sunday, May 29, 2016
1. Whats the relationship between median and percentile?
Median divides the data into 50% on each of its sides, the 5
College of the North Atlantic  Happy ValleyGoose Bay Campus
Stats
MATH 101

Fall 2016
Understanding the ttest as a variance ratio test, and why t2 = F.
ABSTRACT
This unpublished paper provides a rationale for regarding the ttest statistically as a
variance ratio test, on the same basis as Fisher's Ftest used in the analysis of variance.
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
MATH 101

Summer 2013
Meaningful brands from meaningless differentiation: The dependence on irrelev.
Carpenter, Gregory S; Glazer, Rashi; Nakamoto, Kent
JMR, Journal of Marketing Research; Aug 1994; 31, 3; ABI/INFORM Global
pg. 339
Reproduced with permission of the copyright o
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
MATH 101

Summer 2013
Integration
by parts
mcstackTYparts20091
A special rule, integration by parts, is available for integrating products of two functions. This
unit derives and illustrates this rule with a number of examples.
In order to master the techniques explained
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
MATH 101

Summer 2013
maxins.mcd
George Hardgrove
page 1
Creation Date: 6/14/96
Modified on:
MAXWELL DISTRIBUTION OF
GAS MOLECULE VELOCITIES
by
George Hardgrove
Chemistry Department
St. Olaf College
Northfield, MN 55057
Adapted from a document by Flick Coleman, Wellesley Coll
College of the North Atlantic  Happy ValleyGoose Bay Campus
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MATH 101

Summer 2013
3
Integration By Parts
Formula
udv = uv vdu
I. Guidelines for Selecting u and dv:
(There are always exceptions, but these are generally helpful.)
LIATE Choose u to be the function that comes first in this list:
L: Logrithmic Function
I: Inverse Trig
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
MATH 101

Summer 2013
Strategy for using integration by parts
Recall the integration by parts formula:
u dv = uv
v du .
Strategy for using integration by parts
Recall the integration by parts formula:
u dv = uv
v du .
To apply this formula we must choose dv so that we
can in
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
MATH 101

Summer 2013
Answers to sample problems.
1. Integrate by parts.
x sin x dx = x cos x +
cos x dx = x cos x + sin x + c
2. Integrate by parts twice.
x2 ex dx = x2 ex
2 x ex dx = x2 ex 2x ex +
2 ex dx = x2 ex 2x ex + 2 ex + c
3. Sustitute x = sin y .
1 x2 dx =
=
cos y c
College of the North Atlantic  Happy ValleyGoose Bay Campus
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Summer 2013
Mathematical Database
INTRODUCTORY THEORY OF DIFFERENTIATION
This article is written for beginners. No previous knowledge of differentiation is assumed, and
we shall treat the theory of limits and differentiation in a rigorous yet elementary way. If you h
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MATH 101

Summer 2013
CLASS NOTES FOR DISCRETE MATHEMATICS
NOTE ADDED 14 June 2008
These class notes were used for fifteen years in a discrete math class taught at
Case Western Reserve University until I retired in 1999. I am making them available as
a resource to anyone who w
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MATH 101

Summer 2013
Basic Dierentiation  A Refresher
v3. March 2003
www.mathcentre.ac.uk
c mathcentre 2003
Basic Dierentiation  A Refresher
1
Contents
Foreword
Preliminary work
11.
12.
13.
3
Introduction
9.
10.
2
Reminders
5.
6.
7.
8.
2
How to use this booklet
1.
2.
3.
4.
College of the North Atlantic  Happy ValleyGoose Bay Campus
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MATH 101

Summer 2013
QM Handout Gaussian Integration
Gaussian integration is simply integration of the exponential of a quadratic.
We cannot write a simple expression for an indenite integral of this form
but we can nd the exact answer when we integrate from to . The
basic in
College of the North Atlantic  Happy ValleyGoose Bay Campus
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MATH 101

Summer 2013
INTEGRATION BY PARTS THE DI METHOD
This is a short cut to integration by parts and is especially useful when one has to integrate by parts several times. It is a schematic method of the traditional udv method that usually is written as udv = uv vdu in cal
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
MATH 101

Summer 2013
INTEGRATION BY PARTS
Integration by parts is essentially the reverse of product rule.
d
[f (x)g (x)] = f (x)g (x) + g (x)f (x)
dx
f (x)g (x) =
f (x)g (x) dx = f (x)g (x)
f (x)g (x) dx +
g (x)f (x) dx
f (x)g (x) dx
An easier way to remember this is usuall
College of the North Atlantic  Happy ValleyGoose Bay Campus
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MATH 101

Summer 2013
1
Lecture 5  Examples of Integration by parts
and substitution
Many techniques for integration exist, some harder than others.
When guring out which techniques to use against a particular problem, you
usually try the simplest technique rst, and try harde
College of the North Atlantic  Happy ValleyGoose Bay Campus
Stats
MATH 101

Fall 2016
STATISTICS: EXAM 3 REVIEW
On average, nationwide, batteries tend to last 54 months. Another company is interested in knowing if
they have successfully increased the battery life at a level of significance of 0.01. If a random sample of
40 batteries is fou