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Find complete families of polynomial anti-derivatives

Start a set of 8
Step 2 of 16

Change C through all five values. Which information stays fixed?

For F(x) = x cubed plus C, changing C from minus two to two changes F(2) while its derivative at two remains twelve. Row 1: Constant 0, F(2) 8, Derivative at 2 12ConstantF(2)Derivative at 20812
Function value F(2)
8
0

Does C change the derivative?

Year 12, VCE Units 3 & 4, Chapter 7, 2 difficulty levels, No calculatorBuild a worksheet

Anti-derivatives and arbitrary constants

Reverse differentiation and retain the complete family unless a representative is requested. Power-sum preparation follows in 7B

\[\int f(x)\,dx=F(x)+C\]
  1. Reverse the derivative rule For each term with a coefficient and a nonnegative integer power, raise the power by one and divide the coefficient by the new power. Differentiation verifies the result
  2. Keep one arbitrary constant The complete family adds one arbitrary constant to a representative. Constants disappear under differentiation. One chosen representative is not the whole family

Worked example

Find the complete family for the derivative shown below

\[F'(x)=4x^3-6x+5\]
  1. \(\displaystyle F(x)=x^4-3x^2+5x+C\)Increase each power and divide by the new power
  2. \(\displaystyle F'(x)=4x^3-6x+5\)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 differentiation
Omitting the family
✕Give P alone when the whole family is requested✓Give P + C with C arbitrary
Every constant shift has the same derivative

Difficulty levels

introReverse constants and monomials
examRecognise the complete family and distinguish an explicitly requested zero-constant representative

Questions about this topic

Why is there one arbitrary constant?+

Two differentiable functions with the same derivative on a connected interval differ by a constant. Separate disconnected intervals can need independent constants

Where is integrand preparation practised?+

Products, quotients and general power sums belong to 7B. This placement keeps the constant and family concept focus

How can I check an anti-derivative?+

Differentiate every term of the proposed function. The constant differentiates to zero, and the result must match the original integrand

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