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.

- 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.

- 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.

- 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,

- For any real value of p,

Proof: From property 1 we can say that

Also,

From property 2 we can say that

- For a finite number of functions f
_{1}, f_{2}…. f_{n}and the real numbers p_{1}, p_{2}…p_{n},

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