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Partial fractions
Lesson 10 of 75
Review

Partial-fraction modelling

Objective. Use decomposition in recurrence and rate contexts.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Partial-fraction modelling applies decomposition to real contexts such as population growth or circuit rates, where a rational rate function must be integrated to find a quantity over time. Example: if the rate of chemical dilution is dC/dt=10/((t+1)(t+3)), decomposing into simple fractions before integrating gives an explicit formula for concentration C(t) at any time, essential for predicting long-term behaviour. These models often require an initial condition to fix the constant of integration.

Worked example

A rate model gives dV/dt=6/((t+1)(t+2)). Use partial-fraction modelling to find V(t) given V(0)=0.

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Bar chart of the values 6, 1, 2, 0, 0
  1. 1Decompose 6/((t+1)(t+2))=A/(t+1)+B/(t+2), giving A=6 and B=−6
  2. 2Integrate: V(t)=6ln|t+1|−6ln|t+2|+c
  3. 3Apply V(0)=0: 0=6ln1−6ln2+c, so c=6ln2
  4. 4Write the final model combining logarithms

Answer: V(t)=6ln|t+1|−6ln|t+2|+6ln2

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Common mistake. Students integrate the rate function directly without decomposing first, attempting an incorrect standard integral form; always decompose into partial fractions before integrating a rational rate model.

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