
Integral
calculus 

The
definite and indefinite integrals 
The indefinite integral 
Evaluating the indefinite integral example 
The fundamental theorem of
calculus 
The fundamental theorem of differential calculus 






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'. 

Let's
prove that the differentiation of an indefinite integral with respect to
the variable upper limit yields the function
that
is integrated, the integrand. 
Let
write the increment DF
of the difference quotient DF/Dx 

As f(x)
is continuous function in the closed interval 
[x,
Dx]
it takes its maximal value M
and minimal 
value m
at some points, x_{1}
and x_{2},
as is shown in 
the right
figure. 
Therefore,
according to the mean value theorem 




Now,
as Dx
®
0
then x
+ Dx
tends to x,
and since f
is continuous then f
(x_{1})
and f (x_{2})
approach f (x). 
Thus, 


that is, by differentiating the indefinite integral of a
continuous function with respect to
the upper limit x
we get the
function that
is integrated (the integrand). Hence, we also can write 

that
is, by integrating the derivative F' (x),
or the differential dF,
we get the antiderivative F
(x). 
Therefore,
differentiation and integration are inverserelated operations, when
successively performed on the same
continuous function, the function stays unchanged. So, by integrating a
continuous function f
we get a new
function which, when differentiated, leads back to the original function
f. 
This property we use to check the result of integration thus, by differentiating the result of integration we must obtain the integrand
(the function that is integrated). 
Similarly,
differentiation of the indefinite integral with respect to
the lower limit we write as 

so
that, F' (x)dx
= 
f (x). 
Therefore,
the indefinite integral of a continuous function f
(x)
is any of its antiderivatives or primitive functions (whose derivative is
f (x)). 

The fundamental theorem of calculus 
The
theorem that states the relationship between integration and
differentiation, that is, between areas and tangent
lines, is called the fundamental theorem
of calculus 

The fundamental theorem of differential calculus 
If
f (x) is continuous on
closed interval [a,
b] and F
(x)
is defined to be 

then,
F
is differentiable on (a,
b)
such that F' (x)
= f (x) for all x
in (a,
b).
This means that 


Evaluating definite integrals
using indefinite integrals 
To
evaluate the definite integral 

we should find one
primitive function F
(x)
or antiderivative of 

the function f (x),
and since the indefinite integral 

is a primitive
function of f
(x)
then, as two 

primitives
of the same function can differ only by a constant, we can write 

To
find the value of the constant C
that belongs to the lower limit a,
substitute x = a
to both sides of the 
above equality, and
since 

then C
= 
F(a),
so that 


Therefore,
the
definite and indefinite integrals are related by the fundamental theorem
of calculus. 
This
result shows that integration is inverse of differentiation. 

The fundamental theorem of integral calculus 
If
f (x)
is integrable on the interval [a,
b]
and F (x)
is an antiderivative of f
on (a,
b),
then 

The
right side of the above equation we usually write 

so
that, 



Thus,
to evaluate the definite integral we need to find an atiderivative F
of f,
then evaluate F (x)
at x = b
and at x
= a, and calculate the
difference F (b)

F (a). 










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