Integration: Summary
Philippe B. Laval
KSU
September 14, 2005
Abstract
This handout summarizes the various integration techniques. It also
give alternatives for finding definite integrals when an antiderivative cannot be found.
1
Strategy for Integration

5. DIFFERENTIATION
5.1.
The derivative in a point
5.1.1. A rate of change
When a train travels 108 km in a time interval [t1,t2] = [0,8;2,3].of (=90 min = 1,5 h) we
s 108
km / h 72 km / h. .
conclude that its average velocity was v
t 1,5
s2
s
(km)
s1
t
t

3. LIMITS
The concept limit allows us to investigate the behaviour of a function in the close
neighbourhood of a point. The functional value in this point is not important here.
Example 1
(x 1)(x 2)
f(x)
x 1
met
dom f = IR \ 1
3
The function f is not def

7. INFINITE SEQUENCES AND SERIES
7.1. Infinite sequences in IR
7.1.1. Definition
An infinite sequence of real numbers is a mapping of IN ( or IN0 ) in IR
f : IN 0 IR : n f (n) un
Every n corresponds with a real number un. The natural order 1 , 2 , 3 , . o

11. APPLICATIONS OF DEFINITE INTEGRALS
11.1. Surface of a plane area
11.1.1. Surface limited by y = f(x), the x-axis and the lines x = a and x = b
Let us repeat 10.2.4. :
If the function f is continuous on a, b and f 0 for every x a, b then the value of t

4. TRANSCENDENTAL FUNCTIONS
4.1.
Trigonometric functions
The trigonometric functions are important because they are periodic, or repeating, and
therefore model many naturally occurring periodic processes, e.g. a mass attached to a
string, the pendulum, an

8. COMPLEX NUMBERS
8.1. Definition and operations
Imagination was called upon at least three times in constucting real numbers: first the set
of all integers was constructed from the counting numbers, then the set of rational
numbers was constructed from

2. CONTINUITY
2.1. Continuity at a point
Definition: a function f is continuous in a point a if the graph of the functions does not
show a jump in a.
If the function is not continuous in a, it is defined as discontinuous in a.
Examples
y
y
y
f(a)
f(a)
a
a

Name: _ Class: _
Integrating f (ax + b) and Integration by Substitution
Question 1: Find the antiderivative of each function with respect to x:
a)
b)
F ( x) sin( 2 x 3)
F ( x) (3 x 5) 6
Question 2: Integrate with respect to x:
a)
b)
4
(2 x 3) dx
Question

Question 3: What does the second derivative of a function tell you about a function?
The first derivative of a function tells us the rate at which a function changes. If the first
derivative is positive over some interval, then the values on the function

Conics Summary
Academic Skills Advice
Conics are curves that are produced when a double cone is intersected by a plane. The 3
main types of conics are:
Parabola
Ellipse (including the circle which is a special case)
Hyperbola
General Equations of Conics:

Academic Skills Advice
Sketching Curves Summary
When sketching a curve its useful to have an idea of the general shape.
= 2
= 2
= 3 +
= 3 +
Then there are 2 things to think about:
the and intercepts (where does it cross the axes)
the turning point

IB Math HL Integration Practice
Name: _
Integrate each of the following.
1) (a)
(b)
4 x dx
(c)
x
4 x 2 dx
(d)
x
4 x dx
2) (a)
3xe
x 2 5
4
x 2 dx
(b)
dx
3xe
x 5
dx
3) (a)
(b)
sin dx
2 x
3
4) (a)
2
x 6 x 13
3x dx
sin dx
3 x
3
(b)
x 3
x 2 6 x 13 dx
(c)
x
2

Integration by Substitution
Dr. Philippe B. Laval
Kennesaw State University
August 21, 2008
Abstract
This handout contains material on a very important integration method
called integration by substitution. Substitution is to integrals what the
chain rule

9. INDEFINITE INTEGRALS
9.1. Antiderivative
9.1.1. Definition
Let us consider a function f : IR IR: x f ( x).
The function F is an antiderivative of f if F ( x) f ( x) , or dF ( x) f ( x) dx .
Example
Let f : x x 4 .
A few antiderivatives of f are:
x5
F1