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 \(e^x \)
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.
