Antiderivatives
A function whose derivative is f.
Key idea
A function whose derivative is f.
In depth
An antiderivative F of f satisfies F′ = f. It reverses differentiation, but is unique only up to an added constant C, since constants vanish when differentiated.
Worked example
An antiderivative of 3x² is x³ + C. An antiderivative of cos x is sin x + C.
Formula
Real-world uses
- Finding a position from a velocity formula
- Recovering total quantities from rates
Common mistake
Leaving out the + C.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Antiderivatives?
A function whose derivative is f. It is a topic at the advanced level in Calculus.
What is the formula for Antiderivatives?
The key formula is ∫ f(x) dx = F(x) + C.
Where is Antiderivatives used in real life?
Real-world examples: Finding a position from a velocity formula; Recovering total quantities from rates.
What should I learn before Antiderivatives?
Make sure you are comfortable with Newton’s Method, the previous topic in Calculus.
What is a common mistake with Antiderivatives?
Leaving out the + C.
How can I practise Antiderivatives?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Antiderivatives?
Move on to Indefinite Integral Rules next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.