Worksheet: Integration by Partial Fractions of Improper Fractions

In this worksheet, we will practice using partial fractions of rational functions to find the integration of improper fractions.

Q1:

Use partial fractions to evaluate ο„Έ π‘₯ π‘₯ + 2 π‘₯ + 1 π‘₯ 1 0 3 2 d .

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

Q2:

Use partial fractions to evaluate ο„Έ 1 6 π‘₯ 4 π‘₯ βˆ’ 4 π‘₯ + 1 π‘₯ 3 2 d .

  • A 2 π‘₯ + 4 π‘₯ + 3 | 2 π‘₯ βˆ’ 1 | + ( 2 π‘₯ βˆ’ 1 ) + 𝐾 2 βˆ’ 2 l n
  • B π‘₯ + 4 π‘₯ βˆ’ 3 | 2 π‘₯ βˆ’ 1 | βˆ’ ( 2 π‘₯ βˆ’ 1 ) + 𝐾 2 βˆ’ 1 l n
  • C 2 π‘₯ + 4 π‘₯ + | 2 π‘₯ βˆ’ 1 | βˆ’ ( 2 π‘₯ βˆ’ 1 ) + 𝐾 2 βˆ’ 1 l n
  • D 2 π‘₯ + 4 π‘₯ + 3 | 2 π‘₯ βˆ’ 1 | βˆ’ ( 2 π‘₯ βˆ’ 1 ) + 𝐾 2 βˆ’ 1 l n
  • E 2 π‘₯ + 2 π‘₯ βˆ’ 3 | 2 π‘₯ βˆ’ 1 | + ( 2 π‘₯ βˆ’ 1 ) + 𝐾 2 βˆ’ 1 l n

Q3:

Use partial fractions to evaluate ο„Έ π‘₯ π‘₯ βˆ’ 1 π‘₯ 4 2 d .

  • A π‘₯ 3 βˆ’ π‘₯ + 1 2 | | | π‘₯ βˆ’ 1 π‘₯ + 1 | | | + 𝐾 3 l n
  • B 2 π‘₯ 3 + 1 2 | | | π‘₯ βˆ’ 1 π‘₯ + 1 | | | + 𝐾 3 l n
  • C π‘₯ 3 βˆ’ 1 4 | | | π‘₯ βˆ’ 1 π‘₯ + 1 | | | + 𝐾 3 l n
  • D π‘₯ 3 + π‘₯ + 1 2 | | | π‘₯ βˆ’ 1 π‘₯ + 1 | | | + 𝐾 3 l n
  • E π‘₯ 3 + 1 2 | | | π‘₯ βˆ’ 1 π‘₯ + 1 | | | + 𝐾 3 l n

Q4:

Use partial fractions to evaluate ο„Έ 2 π‘₯ βˆ’ 4 π‘₯ βˆ’ π‘₯ βˆ’ 3 π‘₯ βˆ’ 2 π‘₯ βˆ’ 3 π‘₯ 3 2 2 d .

  • A π‘₯ + 2 | π‘₯ βˆ’ 1 | + 3 | π‘₯ βˆ’ 3 | + 𝐾 l n l n
  • B 2 π‘₯ + 2 | π‘₯ βˆ’ 1 | + 3 | π‘₯ βˆ’ 3 | + 𝐾 2 l n l n
  • C π‘₯ + | π‘₯ βˆ’ 1 | + 3 | π‘₯ βˆ’ 3 | + 𝐾 2 l n l n
  • D π‘₯ + 2 | π‘₯ + 1 | + 3 | π‘₯ βˆ’ 3 | + 𝐾 2 l n l n
  • E π‘₯ + 2 | π‘₯ + 1 | + 3 | π‘₯ + 3 | + 𝐾 2 l n l n

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