# 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 ,

then , The function is said to be antiderivative of function .

But as the derivative of constant is zero. Not only , but is also the antiderivative of function where , “c” is any constant.

Or,

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,

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

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:

The integral is denoted by elongated “s” sign ( )

The in the notation is differential of “x” and denotes that integration is to be one with respect to variable “x”.

**Note**:

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

Where “k_{1}” and “k_{2}” are constants.

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