Indefinite Integrals
Learn indefinite integrals for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-05). The integral sign without limits stands for every primitive of a function at once — a whole family of curves that differ only by a vertical shift, recorded with the constant \(C\).
You will learn to use integral notation, integrate powers including negative and fractional ones, split sums and pull out constant multiples, and never drop the constant of integration — the routine fluency every integration topic relies on.
Theory
An indefinite integral has no limits, so its answer is a family \(F(x)+C\). This Year 12 Mathematics Advanced topic (MAV-12-05) applies the integration rules term by term.
\(\displaystyle\int f(x)\,dx=F(x)+C\), where \(F'=f\). Integrate term by term; constants multiply through; always include \(+C\).
The power rule \(\int x^{n}dx=\dfrac{x^{n+1}}{n+1}+C\) also handles negative powers such as \(\int x^{-2}dx=-\dfrac1x+C\).
Method
- Integrate each term with the power rule.
- Keep constant multiples out the front.
- Add \(+C\); check by differentiating.
| \(\ \) | \(=\) | \(5x+C\) |
| \(\ \) | \(=\) | \(x^{4}+2x+C\) |
| \(\ \) | \(=\) | \(-\dfrac1x+C\) |
| \(\ \) | \(=\) | \(2x^{3}+C\) |
Common pitfalls
Frequently asked questions
What is an indefinite integral?
It is an integral with no limits. Its answer is a family of functions F of x plus C, the general antiderivative.
What is the difference between definite and indefinite integrals?
A definite integral has limits and gives a number (an area). An indefinite integral has no limits and gives a family of functions plus C.
How do you integrate a negative power?
Use the power rule: add one to the power and divide. For example the integral of x to the minus 2 is minus 1 over x plus C.
How can you check an integral?
Differentiate your answer; you should get back the original function.