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Worksheet: Integration by Partial Fractions with Nonrepeated Linear Factors

Q1:

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

  • A l n | | π‘₯ π‘₯ + 2 | | + 𝐾
  • B l n √ π‘₯ ( π‘₯ + 2 ) + 𝐾
  • C 1 2 | | | ο„ž π‘₯ π‘₯ + 2 | | | + 𝐾 l n
  • D l n ο„Ÿ | π‘₯ | | π‘₯ + 2 |
  • E 1 2 √ π‘₯ ( π‘₯ + 2 ) + 𝐾 l n

Q2:

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

  • A βˆ’ 2 7 | π‘₯ + 6 | βˆ’ 5 7 | π‘₯ βˆ’ 1 | + 𝐾 l n l n
  • B 2 7 | π‘₯ + 6 | βˆ’ 5 7 | π‘₯ βˆ’ 1 | + 𝐾 l n l n
  • C 2 7 | π‘₯ βˆ’ 1 | + 5 7 | π‘₯ + 6 | + 𝐾 l n l n
  • D 2 7 | π‘₯ + 6 | + 5 7 | π‘₯ βˆ’ 1 | + 𝐾 l n l n
  • E 2 7 1 | π‘₯ βˆ’ 6 | + 5 7 1 | π‘₯ + 1 | + 𝐾 l n l n

Q3:

Use partial fractions to evaluate ο„Έ 1 𝑑 + 𝑑 βˆ’ 2 𝑑 𝑑 3 2 d .

  • A βˆ’ 1 2 | 𝑑 | βˆ’ 1 6 | 𝑑 + 2 | + 1 3 | 𝑑 βˆ’ 1 | + 𝐾 l n l n l n
  • B 1 2 | 𝑑 | + 1 6 | 𝑑 + 2 | + 1 3 | 𝑑 βˆ’ 1 | + 𝐾 l n l n l n
  • C βˆ’ 1 2 | 𝑑 | + 1 6 | 𝑑 + 2 | βˆ’ 1 3 | 𝑑 βˆ’ 1 | + 𝐾 l n l n l n
  • D βˆ’ 1 2 | 𝑑 | + 1 6 | 𝑑 + 2 | + 1 3 | 𝑑 βˆ’ 1 | + 𝐾 l n l n l n
  • E βˆ’ 1 2 | 𝑑 | + 1 6 | 𝑑 βˆ’ 2 | + 1 6 | 𝑑 + 1 | + 𝐾 l n l n l n

Q4:

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

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

Q5:

Use partial fractions to evaluate ο„Έ π‘₯ + 4 π‘₯ ( π‘₯ + 1 ) π‘₯ 1 1 2 d .

  • A 3 2 + 4 1 2 + 3 3 2 l n l n l n
  • B βˆ’ 3 2 + 4 1 2 + 3 3 2 l n l n l n
  • C βˆ’ 3 2 βˆ’ 4 1 2 βˆ’ 3 3 2 l n l n l n
  • D βˆ’ 3 2 βˆ’ 4 1 2 + 3 3 2 l n l n l n
  • E βˆ’ 3 2 βˆ’ 4 1 4 + 3 3 4 l n l n l n