Integral calculus
The definite and indefinite integrals

The fundamental theorem of calculus

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'.
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, x1 and x2, 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 (x1) and f (x2) 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 inverse-related 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).
Functions contents G