Primitive Functions
Learn primitive functions for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-05). Finding a primitive means working backwards: you look for a function with the given derivative, so integration begins by undoing differentiation.
You will learn to find primitives of sums of powers using the reverse power rule, include the constant \(C\), rearrange expressions into a form you can work with, and use an initial condition to find the exact function — the base skill for every later integral.
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.