Calculus

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

If $F(x)=2x^3$, then $F^{\prime }(x)=6x^2$

Thus, we can call $f(x)=2x^3$ the antiderivative of $F^{\prime }(x)=6x^2$

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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 $F^{\prime }(x)=x^3$

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

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

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

Antiderivatives are not unique. For example,

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

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.

Designed by Matthew Cheung. This work is licensed under a Creative Commons Attribution 4.0 International License.