Worksheet: Integration by Partial Fractions of Improper Fractions

In this worksheet, we will practice using partial fractions to find the integrals of rational functions where the degree of the numerator is higher than the degree of the denominator.

Q1:

Use partial fractions to evaluate 𝑥𝑥+2𝑥+1𝑥d.

  • A322ln
  • B3122ln
  • C32+2ln
  • D322ln
  • E312+2ln

Q2:

Use partial fractions to evaluate 16𝑥4𝑥4𝑥+1𝑥d.

  • A2𝑥+2𝑥3|2𝑥1|+(2𝑥1)+𝐾ln
  • B2𝑥+4𝑥+3|2𝑥1|(2𝑥1)+𝐾ln
  • C𝑥+4𝑥3|2𝑥1|(2𝑥1)+𝐾ln
  • D2𝑥+4𝑥+3|2𝑥1|+(2𝑥1)+𝐾ln
  • E2𝑥+4𝑥+|2𝑥1|(2𝑥1)+𝐾ln

Q3:

Use partial fractions to evaluate 𝑥𝑥1𝑥d.

  • A𝑥3+𝑥+12|||𝑥1𝑥+1|||+𝐾ln
  • B𝑥3+12|||𝑥1𝑥+1|||+𝐾ln
  • C𝑥3𝑥+12|||𝑥1𝑥+1|||+𝐾ln
  • D2𝑥3+12|||𝑥1𝑥+1|||+𝐾ln
  • E𝑥314|||𝑥1𝑥+1|||+𝐾ln

Q4:

Use partial fractions to evaluate 2𝑥4𝑥𝑥3𝑥2𝑥3𝑥d.

  • A𝑥+2|𝑥+1|+3|𝑥+3|+𝐾lnln
  • B𝑥+|𝑥1|+3|𝑥3|+𝐾lnln
  • C2𝑥+2|𝑥1|+3|𝑥3|+𝐾lnln
  • D𝑥+2|𝑥1|+3|𝑥3|+𝐾lnln
  • E𝑥+2|𝑥+1|+3|𝑥3|+𝐾lnln

Q5:

Use partial fractions to evaluate 2𝑥2𝑥+1𝑥𝑥𝑥d.

  • A2𝑥|||𝑥1𝑥|||+𝐾ln
  • B𝑥+|||𝑥1𝑥|||+𝐾ln
  • C2𝑥+|||𝑥1𝑥|||+𝐾ln
  • D𝑥+|||𝑥1𝑥|||+𝐾ln
  • E𝑥|||𝑥1𝑥|||+𝐾ln

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