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If a car's speedometer reads a constant 60 mph60\text{ mph}, you know its speed at every second, but can you tell where the car actually started its journey?
In differential calculus, you start with a position function and take the derivative to find velocity. Going backward from a rate of change to the original function is called antidifferentiation.
Formally, a function FF is an antiderivative of ff on an interval if F′(x)=f(x)F'(x) = f(x) for every xx in that interval.
If differentiation has only one exact answer, does running it in reverse also give just one single function?

The Constant of Integration

Consider the functions y=x2y = x^2, y=x2+4y = x^2 + 4, and y=x2−9y = x^2 - 9. When you take the derivative of each, the constant disappears because the derivative of any constant is zero, leaving y′=2xy' = 2x for all three.
Because infinitely many vertical shifts produce the exact same derivative, we represent the entire family of functions by adding an arbitrary constant CC, writing F(x)+CF(x) + C.
How do mathematicians write down this reverse operation in standard calculus notation?

Indefinite Integral Notation

In 1675, Gottfried Wilhelm Leibniz introduced the elongated 'S' symbol (∫\int, representing the Latin word summa) to denote integration.
The expression ∫f(x) dx=F(x)+C\int f(x)\,dx = F(x) + C is called an indefinite integral, where f(x)f(x) is the integrand and dxdx identifies the variable of integration.