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 with coefficient a and a nonnegative integer power, increase the power by one and divide a by the new power. Differentiation verifies the result
- Keep one arbitrary constant The complete family adds an arbitrary constant C to a primitive. 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, choose the expanded polynomial that evaluates to zero at zero. Otherwise give the complete family
Worked example
Find the complete family for this derivative
\[F'(x)=4x^3-6x+5\]
- \(\displaystyle F(x)=x^4-3x^2+5x+C\)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 the primitive plus an arbitrary constant C
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 8: Integration
In this chapter
Approximate area by an endpoint or trapezium ruleEvaluate and transform a definite integralElementary primitives and exact definite integralsFind complete families of polynomial anti-derivativesThe area function and the fundamental theorem of calculusSigned integrals, geometric area and motionFind the area between two curvesAverage value of a functionTotal change and kinematics from a rateDetermine a polynomial from its derivative and a boundary conditionRecover motion from acceleration