The integral

1. H. Dobroshtan
2017

2.  A physicist who knows the velocity of a particle
might wish to know its position at a given time.
 A biologist who knows the rate at which a
bacteria population is increasing might want
to deduce what the size of the population will be
at some future time.
Introduction

3.  In each case, the problem is to find a function F
whose derivative is a known function f.
 If such a function F exists, it is called an
antiderivative of f.
Antiderivatives
Definition
A function F is called an antiderivative of f on
an interval I if F’(x) = f (x) for all x in I.

4. The process of integration reverses the process of
differentiation.
In differentiation, if
( ) 2
2f x x= then ( ) 4f x x′ =
Thus the integral of 4x is,
2
2x
i.e. integration
is the process of moving from f (x)
to f (x).

5.  For instance, let f (x) = x2
.
• It is not difficult to discover an antiderivative of f if
we keep the Power Rule in mind.
• In fact, if F(x) = ⅓ x3
, then F’(x) = x2
= f (x).
Antiderivatives

6.  However, the function G(x) = ⅓ x3
+ 100
also satisfies G’(x) = x2
.
• Therefore, both F and G are antiderivatives of f.
 Indeed, any function of the form H(x)=⅓ x3
+ C,
where C is a constant, is an antiderivative of f.
• The question arises: Are there any others?
Antiderivatives

7.  If F is an antiderivative of f on an interval I, then
the most general antiderivative of f on I is
F(x) + C
where C is an arbitrary constant.
Theorem
Antiderivatives

8.  Going back to the function f (x) = x2
, we see that
the general antiderivative of f is ⅓ x3
+ C.
Antiderivatives

9. Family of Functions
 By assigning specific values to C, we obtain a
family of functions.
• Their graphs are vertical
translates of one another.
• This makes sense, as each
curve must have the same
slope at any given value
of x.

10. Notation for Antiderivatives
 The symbol is traditionally used to
represent the most general an antiderivative of f
on an open interval and is called the indefinite
integral of f .
 Thus, means F’(x) = f (x)
( )f x dx∫
( ) ( )= ∫F x f x dx

11. ( )f x dx∫The expression:
read “the indefinite integral of f with respect to
x,” means to find the set of all antiderivatives of
f.
( )f x dx∫
Integral sign Integrand
x is called the variable
of integration
Indefinite Integral

12. 1. Using Table of Integration Formulas
2. Simplify the Integrand if Possible
Sometimes the use of algebraic manipulation or trigonometric identities will simplify the integrand
and make the method of integration obvious.
3. Look for an Obvious Substitution
Try to find some function in the integrand whose differential also occurs, apart
from a constant factor.
3. Classify the Integrand According to Its Form
Trigonometric functions, Rational functions, Radicals, Integration by parts.
4. Manipulate the integrand.
Algebraic manipulations (perhaps rationalizing the denominator or using trigonometric identities)
may be useful in transforming the integral into an easier form.
5. Relate the problem to previous problems
When you have built up some experience in integration, you may be able to use a method on a
given integral that is similar to a method you have already used on a previous integral. Or you
may even be able to express the given integral in terms of a previous one.
6. Use several methods
Sometimes two or three methods are required to evaluate an integral. The evaluation could involve
several successive substitutions of different types, or it might combine integration by parts with
one or more substitutions.
Strategy for Integration

13. Table of Integration Formulas

14.  For example, we can write
• Thus, we can regard an indefinite integral as representing
an entire family of functions (one antiderivative for each
value of the constant C).
3 3
2 2
because
3 3
x x
x dx C C x
′ 
= + + = ÷
 
∫
Indefinite Integral

15. Every antiderivative F of f must be of the form
F(x) = G(x) + C, where C is a constant.
Example:
2
6 3xdx x C= +∫
Represents every possible
antiderivative of 6x.
Constant of Integration

16. 1
if 1
1
n
n x
x dx C n
n
+
= + ≠ −
+∫
Example:
4
3
4
x
x dx C= +∫
Power Rule for the Indefinite
Integral

18. 1 1
lnx dx dx x C
x
−
= = +∫ ∫
x x
e dx e C= +∫
Indefinite Integral of ex
and bx
ln
x
x b
b dx C
b
= +∫
Power Rule for the Indefinite
Integral

19. Sum and Difference Rules
( )f g dx fdx gdx± = ±∫ ∫ ∫
Example:
( )2 2
x x dx x dx xdx+ = +∫ ∫ ∫
3 2
3 2
x x
C= + +

20. ( ) ( )kf x dx k f x dx=∫ ∫ ( constant)k
4 4
3 3
2 2 2
4 2
x x
x dx x dx C C= = + = +∫ ∫
Constant Multiple Rule
Example:

21. Example - Different Variable
Find the indefinite integral:
27
3 2 6u
e u du
u
 
− + − ÷
 ∫
21
3 7 2 6u
e du du u du du
u
= − + −∫ ∫ ∫ ∫
32
3 7ln 6
3
u
e u u u C= − + − +

22. Position, Velocity, and Acceleration
Derivative Form
If s = s(t) is the position function of an object at
time t, then
Velocity = v = Acceleration = a =
ds
dt
dv
dt
Integral Form
( ) ( )s t v t dt= ∫ ( ) ( )v t a t dt= ∫

23. Integration by Substitution
Method of integration related to chain rule. If u
is a function of x, then we can use the formula
/
f
f dx du
du dx
 
=  ÷
 ∫ ∫

24. Example: Consider the integral:
( )
9
2 3
3 5x x dx+∫
3 2
pick +5, then 3u x du x dx= =
10
10
u
C= +
9
u du∫
( )
10
3
5
10
x
C
+
= +
Sub to get Integrate Back Substitute
2
3
du
dx
x
=
Integration by Substitution

25. 2
Let 5 7 then
10
du
u x dx
x
= − =
Example: Evaluate
( )
3/ 2
1
10 3/ 2
u
C
 
= + ÷
 
( )
3/ 2
2
5 7
15
x
C
−
= +
2
5 7x x dx−∫
2 1/ 21
5 7
10
x x dx u du− =∫ ∫
Pick u,
compute du
Sub in
Sub in
Integrate

26. ( )
3
ln
dx
x x
∫
Let ln thenu x xdu dx= =
( )
3
3
ln
dx
u du
x x
−
=∫ ∫
2
2
u
C
−
= +
−
( )
2
ln
2
x
C
−
= +
−
Example: Evaluate

27. 3
3
2
t
t
e dt
e +∫
3
3
Let +2 then
3
t
t
du
u e dt
e
= =
3
3
1 1
32
t
t
e dt
du
ue
=
+∫ ∫
ln
3
u
C= +
( )3
ln 2
3
t
e
C
+
= +
Example: Evaluate

28. Let f be a continuous function on [a, b]. If F is
any antiderivative of f defined on [a, b], then
the definite integral of f from a to b is defined
by
( ) ( ) ( )
b
a
f x dx F b F a= −∫
The Definite Integral
( )
b
a
f x dx∫ is read “the integral, from a to b of f(x)dx.”

29. In the notation ,
 f (x) is called the integrand.
 a and b are called the limits of integration;
a is the lower limit and b is the upper limit.
 For now, the symbol dx has no meaning by itself; is
all one symbol. The dx simply indicates that the
independent variable is x.
( )
b
a
f x dx∫
Notation

30. The procedure of calculating an integral is called
integration. The definite integral is a
number. It does not depend on x.
Also note that the variable x is a “dummy variable.”
( )
b
a
f x dx∫
( ) ( ) ( )
b b b
a a a
f x dx f t dt f r dr= =∫ ∫ ∫
The Definite Integral

31. Geometric Interpretation
of the Definite Integral
 The Definite Integral As Area
 The Definite Integral As Net Change of Area

32. If f is a positive function defined for a ≤ x ≤ b,
then the definite integral represents the
area under the curve y = f(x) from a to b
( )
b
a
f x dx∫
( )
b
a
A f x dx= ∫
Definite Integral As Area

33. If f is a negative function for a ≤ x ≤ b, then the
area between the curve y = f(x) and the x-axis
from a to b, is the negative of
( ) .
b
a
f x dx∫
Definite Integral As Area
Area from to ( )
b
a
a b f x dx= −∫

34. Consider y = f(x) = 0.5x+6 on the interval [2,6]
whose graph is given below,
Definite Integral As Area
6
2
Find ( )
) by using geometry
) by using the definition
of definite integral
f x dx
a
b
∫

35. Definite Integral As Area
6
2
( ) Area of
Trapezoid
f x dx =∫
Consider y = f(x) = 0.5x+6 on the interval [2,6]
whose graph is given below,

36. ( )
b
a
f x dx =∫ Area of R1 – Area of R2 + Area of R3
a b
R1
R2
R3
If f changes sign on the interval a ≤ x ≤ b, then
definite integral represents the net area, that is, a
difference of areas as indicated below:
Definite Integral as Net Area

37. a b
R1
R2
R3
If f changes sign on the interval a ≤ x ≤ b, and we
need to find the total area between the graph and
the x-axis from a to b, then
Total Area
c d
Total Area  Area of R1 + Area of R2 + Area of R3
Area of R1 ( )
a
c
f x dx= ∫
Area of R2 ( )
d
c
f x dx= −∫
Area of R3 ( )
d
b
f x dx= ∫

38. Example: Use geometry to compute the integral
( )
5
1
1x dx
−
−∫
Area = 2
( )
5
1
1 8 2 6x dx
−
− = − =∫
Area = 8
Area Using Geometry
( ) 1y f x x= = −
–1
5

39. Example: Use an antiderivative to compute the
integral
( )
5
1
1x dx
−
−∫
Area Using Antiderivatives
First, we need an antiderivative of ( ) 1y f x x= = −
( ) 21
( ) 1 . Thus,
2
F x x dx x x C= − = − +∫
( )
5
1
15 3
2 2
1 (5) ( 1) 6C Cx dx F F
−
+ − +
   
− = − − = = ÷  ÷
   ∫

40. Example: Now find the total area bounded by
the curve and the x-axis from
x  –1 to x  5.
Area Using Antiderivatives
( ) 1y f x x= = −
( ) 1y f x x= = −
–1
1 5
R1
R2
Total Area  Area of R1 + Area of R2

41. ( ) 1y f x x= = −
–1
1 5
R1
R2
Area of R1
1
21
1
1
( ) 2
2
x
f x dx x
−
−
 
= − = − = ÷
 
∫
Area of R2
5
25
1
1
( ) 8
2
x
f x dx x
 
= = − = ÷
 
∫
Total Area  2 + 8  10

42. Evaluating the Definite Integral
5
1
1
2 1x dx
x
 
− + ÷
 
∫Example: Calculate
( )
55
2
1 1
1
2 1 lnx dx x x x
x
 
− + = − + ÷
 
∫
( ) ( )2 2
5 ln5 5 1 ln1 1= − + − − +
28 ln5 26.39056= − ≈

43. Substitution for Definite Integrals
( )
1 1/2
2
0
2 3x x dx+∫
2
let 3u x x= +
then
2
du
dx
x
=
( )
1 41/2
2 1/ 2
0 0
2 3x x x dx u du+ =∫ ∫
4
3/ 2
0
2
3
u=
16
3
=
Notice limits change
Example: Calculate

44. Computing Area
Example: Find the area enclosed by the x-axis,
the vertical lines x = 0, x = 2 and the graph of
2
3
0
2x dx∫
Gives the area since 2x3
is
nonnegative on [0, 2].
2
2
3 4
0
0
1
2
2
x dx x=∫ ( ) ( )4 41 1
2 0
2 2
= − 8=
Antiderivative
2
2 .y x=

45. The Definite Integral As a Total
If r(x) is the rate of change of a quantity Q (in units
of Q per unit of x), then the total or accumulated
change of the quantity as x changes from a to b is
given by
Total change in quantity ( )
b
a
Q r x dx= ∫

46. Example: If at time t minutes you are traveling
at a rate of v(t) feet per minute, then the total
distance traveled in feet from minute 2 to minute
10 is given by
10
2
Total change in distance ( )v t dt= ∫
The Definite Integral As a Total