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:

Indefinite Integrals

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:Indefinite Integralsand  Indefinite Integrals

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.

Indefinite Integrals

Then, Indefinite Integrals

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

Indefinite Integrals

Since derivative of any constant function is zero, therefore

Indefinite Integrals

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

Thus, Indefinite Integrals

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:

Indefinite Integrals

Or Indefinite Integrals

On integrating both sides, we get

Indefinite Integrals

where C is any real number.

Indefinite Integrals

Indefinite Integrals

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

Proof: By the first property of integrals we have,

Indefinite Integrals

Also, Indefinite Integrals

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

Indefinite Integrals

  1. For any real value of p,Indefinite Integrals

Proof: From property 1 we can say that Indefinite Integrals

Also,Indefinite Integrals

From property 2 we can say that Indefinite Integrals

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

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