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 right-hand 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 a1 instead of a, such that a1 > a then
Thus, as the point a1 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 F0 that correspond to any chosen area over the interval [a, a1], 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) = x2, 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) = x2 + 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 y-axis by the constant C.
Constant of integration, the arbitrary constant term in the expression of the indefinite integral
Recall that, the constant F0, as part of the expression F (x) = F1(x) + F0, 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) = x2 + 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  - a2.
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) = - x2 + 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 = a3x3 + a1x whose graphs are translated in the direction of the y-axis by  y0 = C.
Let for example  C = - 3 then, the translated cubic  y = (-1/3)x3 + 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 x-axis by x0 = - 4, that is,
we get the cubic explored in the section ' Applications of differentiation - the graph of a function and its derivative'.
Functions contents G
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