College of the North Atlantic  Happy ValleyGoose Bay Campus
101
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
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? F
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 combi
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) +
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 ma
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
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
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
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 w
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?
Media
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 bas
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 Globa
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
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
A
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
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 firs
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
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
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
MATH 101

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 dif
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
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
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
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 th
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
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 ud
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
College of the North Atlantic  Happy ValleyGoose Bay Campus
101
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, yo
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 signific