
Integral
calculus 

The
definite and indefinite integrals 
The indefinite integral 
Integration  inverse of differentiation 
Constant
of integration, the arbitrary constant term in the expression of the
indefinite integral 
Evaluating the indefinite integral example 






The
indefinite
integral 
Assume
the lower limit a
of an integral is fixed and the upper limit is changing. Then, starting
from the fixed
ordinate at a,
also changes the area A
under the curve as its endpoint ordinate moves parallel. 
As
we evaluate the integral over an interval whose righthand endpoint x
changes, the actual value of the area is
a function of x
usually denoted F(x),
so that 

The integral F (x)
is called the indefinite integral of f
(x). Note
that the variable of integration is replaced by the letter t
as x now denotes the upper limit. 



Note also that, the value of the above integral will change whether lower or
upper limit of integration changes. 
So for example, we choose other fixed point a_{1}
instead of a,
such that
a_{1}
> a then 


Thus, as the point a_{1}
can be placed anywhere inside the interval [a,
x], the same
function f (x)
has infinite number
of indefinite integrals that differ only in a constant value F_{0}
that correspond to any chosen area
over the
interval [a,
a_{1}],
as shows the above figure. 

Integration
inverse of differentiation 
We should use the process inverse of differentiation to find function F
(x)
whose derivative is given function f. 
Therefore,
if F (x)
is defined to be the integral of f
(x)
from a
to x
for all x
in [a,
b], written 

then,
f
is derivative of F
at every point of the interval at which f
is continuous. 
This
means that for any x
in (a,
b),
F' (x)
= f
(x). 
Every
such function F(x)
is called an antiderivative or primitive
of a given function f (x).
As, 
dF = F' (x)
dx
= f (x) dx 
the
same process we also can read, how to find the antiderivative F(x)
whose differential f
(x)dx
is given. 
Thus for example, if given 
f
(x)
= 1 then F
(x)
= x, or if f
(x)
= 2x then F(x)
= x^{2}, or if f
(x)
= sin x then F
(x)
= 
cos x and so on. 
But
as well, antiderivatives of these f
(x)
respectively are, 
F
(x)
= x + C,
F (x)
= x^{2} + C,
F (x) = 
cos x + C, and so
on 
regardless
of the value of the constant C. 
Recall
that, according to the mean value theorem, functions that have the same
derivatives at every point of an
interval differ in a constant term only. 
Therefore,
if F (x)
is one of the antiderivative functions of a given f
(x)
then, others are of the form F
(x)
+ C, where C
is any constant. Graphically, they represent the same function
translated in the direction of the yaxis
by the constant C. 

Constant
of integration, the arbitrary constant term in the expression of the
indefinite integral 
Recall
that, the constant F_{0}, as part of the expression
F (x) = F_{1}(x) +
F_{0}, that
comes from the change of the lower limit of integration, as the value in
which differ the indefinite integrals of the same function, does not
include all antiderivatives. 
Thus, for example the expression
F (x) =
x^{2} + C includes
all antiderivatives or primitive
functions of 
f
(x) =
2x since C
can be any positive or negative constant, while the same indefinite
integral when 
evaluated
using the definition 


yields
the negative constant 
a^{2}. 
To
ensure that the indefinite integral coincides with the meaning of
antiderivative or primitive function it is necessary
add an arbitrary constant term to its definition, regardless of the lower
limit a,
so that 

where C
is an arbitrary constant, or written shorter 

thus 



Evaluating
the indefinite integral example 
Example:
Given
f (x) = 
x^{2} + 4, find
its antiderivatives or primitive functions
F (x)
and draw their graphs. 
Solution: Since
an indefinite integral is any function F
(x)
whose derivative is given f
(x),
then 

Therefore,
obtained primitive functions F
(x)
are
represented by all source cubic functions y
= a_{3}x^{3} + a_{1}x
whose
graphs are translated in the direction of the yaxis
by y_{0}
= C. 
Let
for example C = 
3 then, the translated cubic y = (1/3)x^{3}
+ 4x 
3 represents one particular antiderivative
F (x),
as is shown in the figure below (colored blue). 

If,
the graph of F (x)
we now translate, or move horizontally in the direction of xaxis
by x_{0} = 
4, that is, 

we
get the cubic explored in the section ' Applications
of differentiation  the graph of a function and its derivative'. 










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