Trig Integrals
Products of Sines and Cosines
We can sometimes integrate a function of the form sinm x cosn x using an appropriate u-substitution.
We can u-substitution if n or m is odd. The appropriat
Integration By Parts
The Tabular Method
Sometimes it is necessary to apply the method of integration by parts several times before we obtain
an integral we can evaluate. In the case where the integral
Integration By Parts
The Product Rule
Let f and g be functions and suppose that G is an antiderivative for g . This means that G (x) = g (x).
The product rule states that
d
[f (x)G(x)] = f (x)g (x) +
Hyperbolic Functions
Definition
We can express the exponential function ex as the sum of an odd function and an even function:
ex + ex
ex ex
+
.
2
2
The even part of ex is called hyperbolic cosine and
Length of a Plane Curve
Arc Length
Definition Let f be a dierentiable with a continuous derivative f . In this case f is called a
smooth function and the graph y = f (x) is called a smooth curve.
We w
Functions and Integration
Functions Defined by Integrals
Let f be a continuous function. Then by the Fundamental Theorem of Calculus, the function
x
g (x) =
f (t) dt
a
is an antiderivative for f (x).
00
Problem 1. Compute 199 .
A) 1
B) 99
C) 100
D) 100!
E) None of these
Problem 2. For any positive integers n, a, and b with 0 < a < b < n we have
n
n
a < b.
A) True
B) False
Problem 3. Suppose that S
Math 2001 Homework #13 Solutions
1. Section 7.1: #3. For each part below, draw a graph with the described properties.
(a) A graph with three vertices and four edges.
Solution: If we have vertices cfw_
Math 2001 Homework #8 Solutions
1. Section 3.2: 11. Suppose a, b, q , and r are nonzero integers such that a = bq + r. Prove
or disprove each of the following statements.
(a) gcd(a, b) = gcd(b, r)
Sol
Math 2001 Homework #7 Solutions
1. Section 3.1 #2. Let a, b, c, d be nonzero integers. Prove the following implications.
(a) If a|b and a|c, then a2 |bc.
Solution: Suppose a|b and a|c. Then there are
Math 2001 Homework #9 Solutions
1. (a) Roughly how many digits does the number 3893 have?
Solution: Very roughly, we use the fact that 32 = 9 10 and get 3893 3 10446 ,
so we get something slightly les
Math 2001 Homework #10 Solutions
1. Section 4.1: 6ab. For each map below, determine the number of southerly paths from
point A to point B .
Solution: We just have to use the same process as we did in
Math 2001 Homework #11 Solutions
1. Section 4.5: #7. An integer between 100 and 999 inclusive is selected at random. If
n = 100d2 + 10d1 + d0 has digits d2 , d1 , and d0 , nd the probability that:
(a)
Trig Substitutions
Eliminating Radicals
Our goal is to use certain u-substitutions involving trig functions to eliminate the radicals inside an
integral that contains one of the expressions:
a2 x2
x2
Method of Partial Fractions
Partial Fraction Decomposition
Suppose we have a rational function, i.e. a function of the form R(x) = P (x)/Q(x) where P and
Q are polynomials. Under certain conditions we
Problem 1. Let S be a set with three subsets A, B , and C . Suppose that x is
an element of S . How many of the sets
A, B, C, A B, A C, B C, A B C
could contain x? Answer as precisely as possible.
A)
Problem 1. Let n be a positive integer. Compute
n
(1)k
k=0
n
k
=
n
n
n
n
+
+ (1)n
.
0
1
2
n
Problem 2. Let S be a set with n elements. How many ways are there to
partition S into two subsets.
A) 1
B)
Problem 1. Is cfw_1, cfw_2, 3, cfw_5 a partition?
A) Yes
B) No
C) Of what?
Problem 2. How many partitions are there of the empty set?
A) 0
B) 1
C)
D) The answer is not dened.
Problem 3. How many dist
Problem 1. How many of the integers x with 1 x 120 are divisible by at
least one of 2 or 5?
A) 12
B) 60
C) 72
D) 84
E) 96
Problem 2. How many of the integers x with 1 x 120 are divisible by at
least o
Problem 1. Prove by contradiction that the sum of a rational number and an
irrational number cannot be rational.
Problem 2. How many ways are there to rearrange the list (1, 2, 3, 4, 5) such
that each
Problem 1. Let n be a positive integer. Compute
n
(1)k
k=0
n
k
=
n
n
n
n
+
+ (1)n
.
0
1
2
n
Solution. The answer is 0. One way to see this is to remember that (x + y )n =
n
n k nk
and substitute x =
Problem 1. Is cfw_1, cfw_2, 3, cfw_5 a partition?
A) Yes
B) No
C) Of what?
Problem 2. How many partitions are there of the empty set?
A) 0
B) 1
C)
D) The answer is not dened.
Solution. The empty part
00
Problem 1. Compute 199 .
A) 1
B) 99
C) 100
D) 100!
E) None of these
Problem 2. For any positive integers n, a, and b with 0 < a < b < n we have
n
n
a < b.
A) True
B) False
Problem 3. Suppose that S
Problem 1. Draw a picture of congruence modulo 4 as a relation on the integers.
Problem 2. Every relation on the empty set is an equivalence relation.
A) True
B) False
Problem 3. Suppose that Q and R
Differential Equations Summary
We give a summary of our results about dierential equations in the form of a ow chart. We will
assume that rst order dierential equations are either linear or separable
First Order Differential Equations
A dierential equation is an equation involving one or more derivatives of an unknown function.
A function y (x) is a solution of a dierential equation if the equatio
Improper Integrals
We can extend the notion of the denite integral (in which we usually require functions to be
continuous and intervals to be nite) in two ways. One way is to allow integrals over inn
Using Integral Tables
For this section, refer to the integral table on the back inside cover of the text. Some common
integrals can also be found at http:/www.integral-table.com/
Perfect Matches
We ha
Math 2001 Homework #6 Solutions
1. Section 2.1 #11. Suppose m and n are integers. Prove that the following statements
are equivalent.
(a) m2 n2 is even.
(b) m n is even.
(c) m2 + n2 is even.
Solution: