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.

Or,

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

$= f(x) + 0$

$= f(x)$

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”.

Note:

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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