# Indefinite Integrals

An integral which is not having any upper and lower limit.

F(x) is the way function f(x) is integrated and it is represented by:

Where in respect to x the integral of f(x) is on the R.H.S.

primitive or anti-derivative is termed as F(x)

Integrand is termed as f(x)

Integrating agent is termed as dx

The constant of integration is an arbitrary constant termed as C

The variable of integration is termed as x

## Properties of Indefinite Integrals

Let us now look into some properties of indefinite integrals.

1. Differentiation and integration are inverse processes of each other since:and

Where C is any arbitrary constant. Let us now prove this statement.

Proof: Consider a function f such that its anti-derivative is given by F, i.e.

Then,

On differentiating both the sides with respect to x we have,

Since derivative of any constant function is zero, therefore

The derivative of a function f in x is given as f’(x), therefore

Thus,

Where, C is an arbitrary constant called as the constant of integration.

1. Two indefinite integrals having the same derivative have the same family of integrals or curves and therefore they are equivalent.

Consider two functions f and g in x such that:

Or

On integrating both sides, we get

where C is any real number.

1. The integral of sum of two functions is equal to the sum of integrals of the given functions, i.e.,

Proof: By the first property of integrals we have,

Also,

Therefore from equation (1) and (2) we have,

1. For any real value of p,

Proof: From property 1 we can say that

Also,

From property 2 we can say that

1. For a finite number of functions f1, f2…. fn and the real numbers p1, p2…pn,