Antiderivatives ( Indefinite Integrals)

Antiderivatives or Indefinite integrals:

If , “f” is a continuous function defined on an open interval (a,b) ;

Then the function “F” ( function F is capital “f”) is called antiderivative of function “f”, if the derivative of function “f” is function “F” on the interval.

Or , If ,  \frac{d F(x)}{dx} = f(x)

then , The function F(x) is said to be antiderivative of function  f(x).

But as the derivative of constant is zero. Not only  F(x), but  F(x) + c is also the antiderivative of function  f(x) where , “c” is any constant.


 \dfrac{[F(x) + c]}{dx} = \dfrac{d F(x)}{dx} + \dfrac{dc}{dx}

= f(x) + 0

 = f(x)

The converse of above statement and proof is “Any two antiderivatives of a function differs by a constant.

If “F” and “G” be antiderivatives of same function “f” then,

 \dfrac{[F(x) - G(x)]}{dx} = \dfrac{d F(x)}{dx} - \dfrac{G(x)}{dx}

= f(x) - f(x)

= 0

From above proof it follows that there exists a constant “c” such that ,

F(x) - G(x) = c

So what we can conclude is if function “F” is an antiderivative of function “f” , then “F(x) + c” gives all possible antiderivatives of “f” When “c” runs through all possible constants or numbers.

And the function “F(x) + c” is called Antiderivative or Indefinite Integral of function “f”.

As we can see we don’t get a fixed antiderivative , Instead we get a zoo of answers (As “c” is any constant) ; so it is called indefinite integral.

Notation of Antiderivative or Indefinite Integral:

After defining what is Antiderivative of indefinite Integral it is desirable to show Indefinite Integral in notations or mathematically.

If , “f(x)” id derivative of “F(x) + c” or, if “F(x) + c” is the antiderivative of “f(x)” then it can be denoted mathematically as:

 \int f(x) dx = F(x) +c

The integral is denoted by elongated “s” sign (  \int )
The dx in the notation is differential of “x” and denotes that integration is to be one with respect to variable “x”.


One basic property of Indefinite Integral that we can use in most of the calculations of Indefinite integral is:

 \int [k_1 f(x) \pm k_2 g(x)] dx  = k_1 \int f(x) dx \pm k_2 \int g(x) dx

Where “k1” and “k2” are constants.

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