# Calculus

## Antiderivatives

The reverse process of the derivative or a differential is called antidifferentiation.

If $$F(x) = 2x^3$$, then $$F'(x) = 6x^2$$

Thus, we can call $$f(x) = 2x^3$$ the antiderivative of $$F'(x) = 6x^2$$

Let's find the anitderivative $$F(x)$$ such that $$f(x) = x^3$$

We know the power rule for the derivative decreases the power by 1,

$$\therefore$$ $$F(x)$$ must have the power $$x^4$$ and $\frac{d}{dx}x^4 = 4x^3$

Now let's find the coefficient $$a$$ of $$F(x)$$ such that $$F'(x) = x^3$$

If $$a = \frac{1}{4}$$ then $$F(x) = \frac{x^4}{4}$$ and $$F'(x) = x^3 = f(x)$$

$\frac{d}{dx}sinx = cosx,$$$cosx$$ is the antiderivative of $$sinx$$,

and $$e^x$$ is the antiderivative of  $$e^x$$

## What is the difference between and antiderivative and an indefinite integral?

Antiderivatives are not unique. For example,

$\qquad \frac{x^3}{3}, \qquad \frac{x^3}{3}+1, \qquad \frac{x^3}{3}+\pi, \qquad \frac{x^3}{3}-\frac{17}{32},$

are all antiderivatives of $$x^2$$.

The difference between the antiderivatives of a given function is the constant.

$$\therefore$$ we can write the antiderivative as $\int x^2 dx = \frac{x^3}{3} + c$, where $$c$$ is any constant, and call this an indefinite integral.