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If a car's speedometer reads a steady 60 mph60\text{ mph}, you know how fast it is moving — but without a starting landmark, can you pinpoint where on the map the journey began?
Calculus faces this exact challenge when we run differentiation in reverse.

The Two-Way Street

Differentiation takes a function and finds its rate of change, called the derivative. An anti-derivative does the exact opposite: it reconstructs the original function from its rate of change.
In 1675, German mathematician Gottfried Leibniz introduced the elongated letter 'S' symbol, ∫\int, representing integration as a continuous summation of tiny slices dxdx.
What happens when multiple completely different functions produce the exact same derivative?

The Mystery of the Missing Constant

Consider three distinct curves: y=x2y = x^2, y=x2+5y = x^2 + 5, and y=x2−100y = x^2 - 100. Differentiating each gives dydx=2x\frac{dy}{dx} = 2x, because the derivative of any fixed number (a constant) is always zero.
Because differentiation destroys constant values, reversing the process leaves an unknown vertical offset. We capture this ambiguity by writing the constant of integration, denoted as +C+ C.
So, how do we write an algebraic rule to reverse the familiar power rule of differentiation?

The Reverse Power Rule