Antiderivatives

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Indefinite Integral (Antiderivatives)
Definition : Let f(x) be a function. Then the family of all its primitives (or antiderivatives) is called the indefinite integral of f(x) and is denoted by

$\int_{}^{} f(x) dx$

OR

A function 'F' is an antiderivative of 'f' when F '(x) = f(x) for all x in the domain of f.

The symbol $\int_{}^{} f(x) dx$ is read as 'integration of f(x) with respect to x'.

Thus, $\frac{\text{d}}{\text{d}x}(g(x) +c) = f(x) \Leftrightarrow \int_{ }^{ } f(x)dx = g(x) + c$

Where g(x) is the antiderivative of f(x) and 'c' is any constant known as the constant of integration .
Here $\int_{}^{}$ integral sign and f(x) is the integrand, x is the variable of integration and dx is the differential of x.

Note: Antiderivative are not unique. A given function can have many antiderivatives because of the value of constant of integration.
For example, antiderivative of $x^{3}$ is

$\frac{x^{4}}{4}$ , $\frac{x^{4}}{4} + 2$, $\frac{x^{4}}{4} - 3$

In general, we can write it as
$\frac{x^{4}}{4} + c$

Proof on Antiderivatives

If F is an antiderivative of f for all x in the domain of f then G is an antiderivative of f on the same domain of f if and only if G is of the form
G(x)= F(x) + c, where c is a constant.
Proof : If G(x) = F(x) + c and F '(x) = f(x) and 'c' is a constant then
G'(x) = $\frac{\text{d}}{\text{d}x}[F(x) +c]$
= F' (x) + 0
G'(x) = f(x)