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Lesson: Integration by Partial Fractions with Nonrepeated Irreducible Quadratic Factors

Worksheet • 3 Questions

Q1:

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

  • A l n l n l n | π‘₯ | βˆ’ 1 3 | π‘₯ + 1 | βˆ’ 1 3 | | π‘₯ βˆ’ π‘₯ + 1 | | + 𝐾 2
  • B l n l n l n | π‘₯ | + 2 3 | π‘₯ + 1 | + 2 3 | | π‘₯ βˆ’ π‘₯ + 1 | | + 𝐾 2
  • C l n l n l n | π‘₯ | βˆ’ 2 3 | π‘₯ + 1 | βˆ’ 2 3 | | π‘₯ βˆ’ π‘₯ + 1 | | + 𝐾 2
  • D l n l n l n | π‘₯ | βˆ’ 1 6 | π‘₯ + 1 | βˆ’ 1 6 | | π‘₯ βˆ’ π‘₯ + 1 | | + 𝐾 2
  • E l n l n l n | π‘₯ | + 1 3 | π‘₯ + 1 | + 1 3 | | π‘₯ βˆ’ π‘₯ + 1 | | + 𝐾 2

Q2:

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

  • A 1 2 π‘₯ + 1 4 | π‘₯ βˆ’ 1 | βˆ’ 1 4 | π‘₯ + 1 | + 𝐾 t a n l n l n βˆ’ 1
  • B 1 2 π‘₯ βˆ’ 1 2 | π‘₯ βˆ’ 1 | βˆ’ 1 4 | π‘₯ + 1 | + 𝐾 t a n l n l n βˆ’ 1
  • C 1 4 π‘₯ + 1 4 | π‘₯ βˆ’ 1 | βˆ’ 1 4 | π‘₯ + 1 | + 𝐾 t a n l n l n βˆ’ 1
  • D 1 2 π‘₯ + 1 4 | π‘₯ βˆ’ 1 | + 1 4 | π‘₯ + 1 | + 𝐾 t a n l n l n βˆ’ 1
  • E 1 2 π‘₯ + 1 2 | π‘₯ βˆ’ 1 | βˆ’ 1 2 | π‘₯ + 1 | + 𝐾 t a n l n l n βˆ’ 1

Q3:

Use partial fractions to evaluate ο„Έ 3 𝑑 + 𝑑 + 4 𝑑 + 𝑑 𝑑 √ 3 1 2 3 d .

  • A l n ο€Ώ 9 √ 2  + πœ‹ 1 2
  • B l n ο€Ώ 9 √ 2  + πœ‹ 6
  • C l n ο€Ό 9 4  + πœ‹ 1 2
  • D l n ο€Ό 9 2  + πœ‹ 1 2
  • E l n ο€Ώ 1 8 √ 2  + πœ‹ 1 2
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