Polynomial anti-derivatives
Reverse the power rule, preserve signs and include the required constant convention
\[ax^n\longmapsto\frac{a}{n+1}x^{n+1}+C\]
- Reverse the derivative rule For each term axⁿ with a nonnegative integer n, raise the power to n+1 and divide a by n+1. Differentiation verifies the result
- Keep one arbitrary constant The complete family is P+C. Constants disappear under differentiation. One chosen representative is not the whole family
- Prepare a polynomial first Expand a product or divide every numerator term by the monomial denominator. Keep the original domain when simplifying a quotient
- Distinguish a specified representative When zero constant term is explicitly requested, use P(0)=0 for the expanded polynomial. Otherwise give the complete family
Worked example
Find the complete family whose derivative is 4x³ − 6x + 5
\[F'(x)=4x^3-6x+5\]
- \(\displaystyle F(x)=x^4-3x^2+5x+C\)Increase each power and divide by the new power
- \(\displaystyle F'(x)=4x^3-6x+5\)Differentiate to verify every term and the disappearing constant
Common mistakes
Repeating differentiation
✕Lower the exponent again✓Increase the exponent by one and divide
Anti-differentiation reverses differentiationOmitting the family
✕Give P alone when the whole family is requested✓Give P + C with C arbitrary
Every constant shift has the same derivativeDifficulty levels
introConstants and monomials
standardSparse polynomial sums
challengeProducts and monomial quotients with a connected domain
examComplete families and explicitly specified zero-constant representatives
Questions about this topic
Which powers occur in these polynomial exercises?+
The integrands use nonnegative integer powers. Negative and fractional powers are outside these polynomial exercises
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 7: Antiderivatives
In this chapter
7A Antiderivatives and arbitrary constants: Find complete families of polynomial anti-derivatives7B Power rule and integrand preparation: Find complete families of polynomial anti-derivatives7B Power rule and integrand preparation: Recognise equivalent polynomial formsSimplify and combine algebraic fractionsAnti-differentiate a power of a linear function7C Integrating powers of linear functions: Recognise equivalent polynomial forms7C Integrating powers of linear functions: Solve a linear equationAnti-differentiate exponential and circular functionsAnti-differentiate by recognition from a supplied derivativeDetermine a polynomial from its derivative and a boundary conditionInfer primitive curves from a gradient functionReconstruct quantities from their ratesEvaluate and determine function rules7E Antiderivative families and applications: Solve a linear equationRound to decimal places and significant figuresScale a proportional quantity with the unitary method