Key takeaway: An antiderivative of a function g is a function G whose derivative is g, and because constants vanish when differentiated, every function has a whole family of antiderivatives.
Grade 12 – AP Calculus ABAP Calculus AB Unit 6Calculus
Definition of Antiderivative
An antiderivative undoes a derivative. If differentiating \(G\) produces \(g\), then \(G\) is an antiderivative of \(g\). Finding one is like recovering a position from a speed: the speed alone does not say where you began.
That missing starting value is the constant \(C\). Any two answers for the same function differ by a constant, so their graphs are vertical shifts of one another. An extra piece of information, such as a known value, fixes the constant.
Formula and notation
The family of all antiderivatives of \(g\) is written with the integral sign:
\[ \int g(x)\,dx=G(x)+C,\qquad\text{where }G^{\prime}(x)=g(x). \]
For powers, \(\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\) when \(n\neq -1\).
Key facts
- Check a candidate by differentiating it; the result must match the original function.
- Two answers for one function differ only by an added constant.
- The power rule runs backward: raise the exponent by one, then divide by the new exponent.
- A condition like G(2) = 40 selects the single curve from the family.
Antiderivative example
Find the function \(G\) whose derivative is \(g(x)=8x^3-6x+5\) and which satisfies \(G(2)=40\).
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Reverse each term: \(G(x)=2x^4-3x^2+5x+C\).
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Use the condition: \(G(2)=32-12+10+C=30+C\), and this must equal 40, so \(C=10\).
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Check by differentiating: \(8x^3-6x+5\), which matches \(g\).
The required function is \(G(x)=2x^4-3x^2+5x+10\).
Common mistake
Watch out
A student writes \(\int x^3\,dx=3x^2+C\), which is the derivative of \(x^3\) rather than its reverse.
Why it is wrong
Going backward moves the opposite way: raise the power to 4 and divide by 4, giving \(\frac{x^4}{4}+C\). Differentiating that returns \(x^3\), which confirms it. The check by differentiation catches this slip every time.
Related terms and lessons
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