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Lesson: Integration by Partial Fractions with Repeated Irreducible Quadratic Factors

Worksheet • 6 Questions

Q1:

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

  • A t a n βˆ’ 1 2 2 2 πœƒ + 2 πœƒ + 1 βˆ’ 1 4 ( πœƒ + 1 ) + 𝐾
  • B t a n βˆ’ 1 2 2 2 πœƒ + 2 πœƒ + 1 + 1 4 ( πœƒ + 1 ) + 𝐾
  • C t a n πœƒ + 2 πœƒ + 1 βˆ’ 1 4 ( πœƒ + 1 ) + 𝐾 2 2 2
  • D t a n βˆ’ 1 2 2 2 πœƒ + 2 πœƒ + 1 βˆ’ 1 ( πœƒ + 1 ) + 𝐾
  • E t a n βˆ’ 1 2 2 2 πœƒ + 1 πœƒ + 1 βˆ’ 1 4 ( πœƒ + 1 ) + 𝐾

Q2:

Use partial fractions to evaluate ο„Έ 𝑠 + 8 1 𝑠 ( 𝑠 + 9 ) 𝑠 4 2 2 d .

  • A l n | 𝑠 | + 9 ( 𝑠 + 9 ) + 𝐾 2
  • B l n | 𝑠 | + 9 ( 𝑠 + 9 ) + 𝐾 2 2
  • C l n | 𝑠 | βˆ’ 9 ( 𝑠 + 9 ) + 𝐾 2
  • D l n | 𝑠 | βˆ’ 1 8 ( 𝑠 + 9 ) + 𝐾 2
  • E l n | 𝑠 | + 1 8 ( 𝑠 + 9 ) + 𝐾 2

Q3:

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

  • A t a n βˆ’ 1 βˆ’ 1 βˆ’ 2 𝑠 + ( 𝑠 βˆ’ 1 ) βˆ’ ( 𝑠 βˆ’ 1 ) + 𝐾
  • B t a n βˆ’ 1 βˆ’ 1 βˆ’ 2 𝑠 + ( 𝑠 + 1 ) + ( 𝑠 + 1 ) + 𝐾
  • C t a n βˆ’ 1 βˆ’ 1 2 βˆ’ 2 𝑠 βˆ’ ( 𝑠 βˆ’ 1 ) βˆ’ ο€Ή 𝑠 + 1  + 𝐾
  • D t a n βˆ’ 1 βˆ’ 1 βˆ’ 2 𝑠 + ( 𝑠 βˆ’ 1 ) + ( 𝑠 βˆ’ 1 ) + 𝐾
  • E t a n βˆ’ 1 βˆ’ 2 βˆ’ 3 𝑠 βˆ’ ( 𝑠 βˆ’ 1 ) βˆ’ ( 𝑠 βˆ’ 1 ) + 𝐾
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