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 a nonnegative integer power, raise the power by one and divide the coefficient by the new power. Differentiation verifies the result
- Keep one arbitrary constant Add an arbitrary constant to a chosen primitive to obtain the complete family. 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 of anti-derivatives
\[F'(x)=4x^3-6x+5\]
- \(\displaystyle F(x)=x^4-3x^2+5x+C\)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 11, VCE Units 1 & 2, Mathematical Methods, chapter 14: Anti-differentiation and integrals