Every time you integrate a function, you're undoing differentiation—but since the derivative of any constant is zero, you unlock infinitely many antiderivatives, all differing by a vertical shift. Drag the slider to explore how the constant C translates your antiderivative curve up and down.
The power rule for integration states that the integral of x to the power n equals x to the power n+1 divided by n+1, plus C—but only when n is not equal to negative one. This rule reverses the power rule for differentiation: since differentiating x to the power n+1 brings the exponent down and reduces it by one, integrating x to the power n does the opposite, increasing the exponent by one and dividing by the new exponent. Constant multiples can be factored outside the integral sign, and the integral of a sum equals the sum of the integrals, so you can integrate polynomials term by term. Every antiderivative includes the constant C because differentiating any constant gives zero, meaning infinite vertical translations of the same curve are all valid antiderivatives.