Antiderivative
A function F whose derivative is f. Because constants vanish when you differentiate, every antiderivative comes with + C.
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Analogy
Rewinding a film: the derivative tells you the speed, the antiderivative rebuilds the journey.
Example
Try F(x) = x^3/3.Check: F'(x) = 3x^2/3 = x^2.So the integral of x^2 dx = x^3/3 + C.C is any constant.
In practice
Used to recover position from velocity or total cost from a rate.
Practice
Flashcards
- What is the integral of x^2 dx?x^3/3 + C.
Questions people ask
Why do we add + C to an indefinite integral?
Many functions share the same derivative, because constants disappear when you differentiate. x squared, x squared plus 5 and x squared minus 3 all give 2x. The C stands for every possible constant, so the answer covers the whole family of antiderivatives.
How can I check an antiderivative is right?
Differentiate your answer. If you get the original function back, the antiderivative is correct. For example, if you claim the integral of x squared is x cubed over 3, differentiating gives x squared. This quick check catches most mistakes, including a forgotten power or a wrong denominator.