1. Calculus 2 (Math 112):
Integration Concepts & Formulas
PART 1
Engr. Ma. Leona Maye B. Pepito
Mechanical Engineering Department
UNIVERSITY OF SCIENCE AND TECHNOLOGY
OF SOUTHERN PHILIPPINES
Alubijid | Cagayan de Oro | Claveria | Jasaan | Oroquieta | Panaon
2. Course Outcome (CO)
CO1:Properly carry out integration through the
use of the fundamental formulas and the various
techniques of integration for both single and
multiple integrals.
3. Intended Learning Outcome
• Familiarize integration concept, integration
formulas and to countercheck anti-
differentiation by its inverse problem
6. You would have noticed that all the functions have the same
derivative of 6x, because when we differentiate the constant term
we obtain zero.
Hence, when we reverse the process, we have no idea what the
original constant term might have been.
So we include in our response an unknown constant, c, called the
arbitrary constant of integration.
The integral of 6x then is 3x2 + c.
7. 1.1. Anti-Differentiation
• In differentiation, the differential coefficient
��
��
indicates that a
function of x is being differentiated with respect to x, the dx
indicating that it is “with respect to x”.
• In integration, the variable of integration is shown by adding
d(the variable) after the function to be integrated. When we
want to integrate a function, we use a special notation:
� � ��.
8. Differentiation
Integration
The process of finding
a derivative The process of finding
the antiderivative
Symbols:
Symbols:
)
(
'
,
'
, x
f
y
dx
dy
dx
x
f )
(
Integral
Integrand
Tells us the
variable of
integration
9.
10. dx
x
f )
( is the indefinite integral of f(x) with respect to x.
Each function has more than one antiderivative (actually infinitely many)
2
3
2
3
2
3
2
3
3
58
3
4
3
6
3
x
x
x
x
x
x
x
x
Derivative of:
The
antiderivatives
vary by a
constant!
12. Note that along with the integral sign ( �� , there is a term of the form dx,
which must always be written, and which indicates the variable involved, in
our example x.
• We say that 4x is integrated with respect x, i.e: �� ��
• The function being integrated is called the integrand.
• Technically, integrals of this type are called indefinite integrals, to
distinguish them from definite integrals, which we will deal with later.
When you are required to evaluate an indefinite integral, your answer must
always include a constant of integration.; i.e:
�� �� = 2�2
+ c; where c ∈ ℝ
13. Definition of Anti-derivative
• Formally, we define the anti-derivative as: If
f(x) is a continuous function and F(x) is the
function whose derivative is f(x), i.e.: �′
(x) =
f(x) , then:
� � �� = F(x) + c; where c is any arbitrary
constant.
14. 1.2 Simple Power Formula
The general solution of integrals of the form
���
dx , where k and n are constants is given
by:
���
dx =
���+�
�+�
+ c ; where � ≠ − � and c ∈ ℝ
15. Ready to use integrals
Function, f (x) Indefinite integral � � ��
f (x)= k , where k is a
constant
��� = kx + c ; where c ∈ ℝ
f (x)= x ��� =
�
�
��
+ c; where c ∈ ℝ
f (x) = x2
��
�� =
�
�
��+ c; where c ∈ ℝ
f(x) = axn
����� =
���+�
�+�
+ c; where c ∈ ℝ
f (x) = �−� =
�
�
�
�
dx = ln � + c; where c ∈ ℝ
f (x) = kanx
����
dx =
����
�.���
+ c ; where c ∈ ℝ ;
a > 0 and a ≠ 1
16. Worked Examples: Indefinite
integrals
a) �7
dx
=
��+�
�+�
+ c ; where c∈ ℝ
=
��
�
+ c ; where c∈ ℝ
Compare �7
dx with
����� =
���+�
�+�
+ c,
Then: a = 1; n = 7 ; n + 1 = 8
17. Worked Examples: Indefinite
integrals
b) 2
�2du
= 2�−2
du
= 2 �−2
du
= 2 (
�−2+1
−2+1
) + c ; where c∈ ℝ
=
−�
�
+ c ; where c∈ ℝ
Note: in this example, we
are integrating with
respect to u.
Explain the method used
18. If f ‘ (x) =
Then f(x) =
If f ‘ (x) =
Then f(x) =
If f ‘ (x) =
Then f(x) =
If f ‘ (x) =
Then f(x) =
3
2
9
4 2
3
x
x
x
x
x
x
x 3
3 2
3
4
2
3
x
x
x
x
x 2
2
3
3
2 2
2
3
5
4
3
2
3 2
x
x
x
x
x
x
x 5
2
3
5
3 2
3
5
5
3
4 2
x
x
x
x
x 5
2
3
3
4 2
3
19. 1.3. Simple Trigonometric Functions
C
a
ax
dx
ax
cos
)
sin(
C
a
ax
dx
ax
sin
)
cos(
C
a
ax
dx
ax
tan
)
(
sec2
C
a
ax
dx
ax
cot
)
(
csc2
C
a
ax
dx
ax
ax
sec
)
tan
(sec
C
a
ax
dx
ax
ax
csc
)
cot
(csc
20. Find:
dx
x5
dx
x
1
xdx
2
sin
dx
x
2
cos
C
x
6
6
C
x
C
x
dx
x
2
2 2
1
2
1
C
x
2
2
cos
C
x
C
x
2
sin
2
2
1
2
sin
You can always check
your answer by
differentiating!
21. Basic Integration Rules
dx
x
f
k
dx
x
kf )
(
)
(
dx
x
g
dx
x
f
dx
x
g
x
f )
(
)
(
)
(
)
(
22. Evaluate:
dx
x
x
x 4
6
5 2
3
dx
dx
x
dx
x
dx
x 4
6
5 2
3
dx
dx
x
dx
x
dx
x 4
6
5 2
3
C
x
C
x
C
x
C
x
4
2
6
3
4
5
2
3
4
C
x
x
x
x
4
3
3
4
5 2
3
4
C represents
any constant
28. Important notes: The general solution for an Indefinite Integral
C
x
F
dx
x
f
)
(
)
(
Where c is a constant
You will lose points if
you forget dx or + C!!!
