Primitive Functions
Primitive functions reverse differentiation: a primitive (anti-derivative) \(F\) satisfies \(F'(x)=f(x)\), and \(\displaystyle\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C\) for \(n\neq-1\).
Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Integral calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
Theory
A primitive (antiderivative) reverses differentiation. This Year 12 Mathematics Advanced topic (MAV-12-05) uses the power rule \(\int x^{n}dx=\dfrac{x^{n+1}}{n+1}+C\) and finds the constant from a known point.
A primitive of \(f\) is a function whose derivative is \(f\). Because the derivative of a constant is \(0\), every primitive includes \(+C\).
Power rule: \(\int x^{n}dx=\dfrac{x^{n+1}}{n+1}+C\) for \(n\neq-1\); a constant gives \(\int k\,dx=kx+C\). A known point lets you find \(C\).
Method
- Add one to each power.
- Divide by the new power.
- Add \(+C\), and use a given point to find it if asked.
| \(\ \) | \(=\) | \(\dfrac{x^{4}}{4}+C\) |
| \(\ \) | \(=\) | \(x^{2}+3x+C\) |
| \(\ \) | \(=\) | \(\dfrac{x^{3}}{3}-2x^{2}+x+C\) |
| \(f(x)\) | \(=\) | \(3x^{2}+C\) |
| \(f(0)=2\) | \(\Rightarrow\) | \(C=2\) |
| \(f(x)\) | \(=\) | \(3x^{2}+2\) |
Common pitfalls
Frequently asked questions
What is a primitive function?
A primitive, or antiderivative, is a function whose derivative is the given function. Finding it is the reverse of differentiating.
What is the power rule for integration?
Add one to the power and divide by the new power: the integral of x to the n is x to the n plus 1, over n plus 1, plus C, provided n is not minus 1.
Why do you add plus C?
Because differentiating a constant gives zero, so any constant could have been there. The plus C represents every possible constant.
How do you find the constant of integration?
Substitute a known point of the function into your answer and solve for C.