
Integral
calculus 

The indefinite integral 
Evaluating the indefinite integral example 






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)). 










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