Indefinite Integrals
Indefinite integrals give the family of primitives \(\displaystyle\int f(x)\,dx=F(x)+C\), always including the constant of integration \(C\).
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
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.