If a function f is differentiable in an interval I, i.e., its derivative f 'exists at each point of I, then a natural question arises that given f 'at each point of I, can we determine the function? The functions that could possibly have given function as a derivative are called anti derivatives (or primitive) of the function.

Further, the formula that gives all these anti derivatives is called the indefinite integral of the function and such process of finding anti derivatives is called integration. This is the basic idea about the integrals.

Problems Arise in Integrals while Doing

Based on the idea of an integral, such type of problems arises in many practical situations. For instance, if we know the instantaneous velocity of an object at any instant, then there arises a natural question, i.e., can we determine the position of the object at any instant? There are several such practical and theoretical situations where the process of integration is involved. The development of integral calculus arises out of the efforts of solving the problems based on the idea of the following types:

(a) the problem of finding a function whenever its derivative is given,

(b) the problem of finding the area bounded by the graph of a function under certain conditions.

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These two problems lead to the two forms of the integrals, e.g., indefinite and definite integrals, which together constitute the Integral Calculus which together constitute the Integral Calculus.

There is a connection depend on the idea, known as the Fundamental Theorem of Calculus, between indefinite integral and definite integral which makes the definite integral as a practical tool for science and engineering. The definite integral is also used to solve many interesting problems from various disciplines like economics, finance and probability.

Integration as an Inverse Process of Differentiation

Integration is the way of inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and asked to find its primitive, i.e., the original function. Such a process is called integration or anti differentiation.

Let us consider the following examples:

We know that

(d/dx) sin x = cos x               - (1)

In this operation (1), the trigonometric function sine with the variable x is differentiated as the cosine function. This is the anti-derivative function while we work with integrals of trigonometric function cosine  it is integrated as to sine function. These forms worked with different basic functions as we said are the ideas of an integral.

Introduction to properties of the integral and the average value:

Integration is one of the essential notions of mathematics. Let us consider a real variable x and its function f in the interval [a, b], where its definite integral is given by,

b
? f(x) dx
a

If f is said to be a function with the variable x in an interval [a, b] and if the anti-derivative of that function is known then the integral is given by

b
? f(x) dx = F(b) – F(a)
a

Properties of the Integral and the Average Value:

In this article we are going to see about the properties of integrals and average value.

Additive properties:

The additive property of the integral is given by

b               c                c
? f(x) dx + ? f(x) dx = ?  f(x) dx
a               b               a

If the upper and the lower limit of an integral is same then its value is equal to zero,

a
? f(x) dx = 0
a

If the limits of integral changes then it also changes in sign,

b                  a
? f(x) dx = - ? f(x) dx
a                 b

Property of scaling by a constant:

b                     b
? c f(x) dx = c ? f(x) dx
a                      a

Here c is constant, so take that out of the integration.

Integral of a sum:

The sum of the integrals can be applied separately,

b                           b                b
? f(x) + g(x) dx  = ? f(x) dx + ? g(x) dx
a                         a               a

Integral inequalities:
b
If f(x) =0, and if a<b, then ? f(x) dx =0
a

b                b
If f(x) =g(x) and a<b then ? f(x) dx = ? g(x) dx
a               a

Properties of the Integral and the Average Value-function in an Interval:

As we know already to calculate the average value of some finite set which is given by a1, a2, a3,.., an is given by

a1+ a2+ a3+,.., +an
an

Let us consider the average value of a function given by

f(x1) +f(x2) + f(x3) + …+ f (xn)
n

If we multiply by ?x and divide by ?x then we get

f(x1) +f(x2) + f(x3) + …+ f (xn) ?x
n?x

Here ?x= (b-a)/n

n?x = b-a

If we apply in the above equation we get,

f(x1) +f(x2) + f(x3) + …+ f (xn) ?x
b-a

Thus the average value is given by

n
Average ˜ (1/(b-a))? f(xi) ?x
i=1