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.

  • A 3 2 2 l n
  • B 3 2 + 2 l n
  • C 3 1 2 2 l n
  • D 3 2 2 l n
  • E 3 1 2 + 2 l n

Q2:

Use partial fractions to evaluate 16𝑥4𝑥4𝑥+1𝑥d.

  • A 2 𝑥 + 2 𝑥 3 | 2 𝑥 1 | + ( 2 𝑥 1 ) + 𝐾 l n
  • B 𝑥 + 4 𝑥 3 | 2 𝑥 1 | ( 2 𝑥 1 ) + 𝐾 l n
  • C 2 𝑥 + 4 𝑥 + 3 | 2 𝑥 1 | ( 2 𝑥 1 ) + 𝐾 l n
  • D 2 𝑥 + 4 𝑥 + | 2 𝑥 1 | ( 2 𝑥 1 ) + 𝐾 l n
  • E 2 𝑥 + 4 𝑥 + 3 | 2 𝑥 1 | + ( 2 𝑥 1 ) + 𝐾 l n

Q3:

Use partial fractions to evaluate 𝑥𝑥1𝑥d.

  • A 2 𝑥 3 + 1 2 | | | 𝑥 1 𝑥 + 1 | | | + 𝐾 l n
  • B 𝑥 3 + 1 2 | | | 𝑥 1 𝑥 + 1 | | | + 𝐾 l n
  • C 𝑥 3 + 𝑥 + 1 2 | | | 𝑥 1 𝑥 + 1 | | | + 𝐾 l n
  • D 𝑥 3 𝑥 + 1 2 | | | 𝑥 1 𝑥 + 1 | | | + 𝐾 l n
  • E 𝑥 3 1 4 | | | 𝑥 1 𝑥 + 1 | | | + 𝐾 l n

Q4:

Use partial fractions to evaluate 2𝑥4𝑥𝑥3𝑥2𝑥3𝑥d.

  • A 𝑥 + 2 | 𝑥 + 1 | + 3 | 𝑥 3 | + 𝐾 l n l n
  • B 𝑥 + | 𝑥 1 | + 3 | 𝑥 3 | + 𝐾 l n l n
  • C 𝑥 + 2 | 𝑥 1 | + 3 | 𝑥 3 | + 𝐾 l n l n
  • D 𝑥 + 2 | 𝑥 + 1 | + 3 | 𝑥 + 3 | + 𝐾 l n l n
  • E 2 𝑥 + 2 | 𝑥 1 | + 3 | 𝑥 3 | + 𝐾 l n l n

Q5:

Use partial fractions to evaluate 2𝑥2𝑥+1𝑥𝑥𝑥d.

  • A 2 𝑥 | | | 𝑥 1 𝑥 | | | + 𝐾 l n
  • B 𝑥 | | | 𝑥 1 𝑥 | | | + 𝐾 l n
  • C 2 𝑥 + | | | 𝑥 1 𝑥 | | | + 𝐾 l n
  • D 𝑥 + | | | 𝑥 1 𝑥 | | | + 𝐾 l n
  • E 𝑥 + | | | 𝑥 1 𝑥 | | | + 𝐾 l n

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