Worksheet: Integration by Partial Fractions with Linear Factors

In this worksheet, we will practice using partial fractions for rational functions with linear factors to evaluate their integration.

Q1:

Use partial fractions to evaluate 𝑥 + 4 𝑥 + 1 ( 𝑥 1 ) ( 𝑥 + 1 ) ( 𝑥 + 3 ) 𝑥 2 d .

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

Q2:

Use partial fractions to evaluate 6 𝑥 + 7 ( 𝑥 + 2 ) 𝑥 2 d .

  • A 6 | 𝑥 + 2 | + 5 ( 𝑥 + 2 ) + 𝐾 l n 3
  • B 6 | 𝑥 + 2 | + 5 | 𝑥 + 2 | + 𝐾 l n l n
  • C 6 | 𝑥 + 2 | 5 ( 𝑥 + 2 ) + 𝐾 l n 1
  • D 6 | 𝑥 + 2 | + 5 ( 𝑥 + 2 ) + 𝐾 l n 1
  • E 3 | 𝑥 + 2 | 4 ( 𝑥 + 2 ) + 𝐾 l n 1

Q3:

Use partial fractions to evaluate 1 ( 𝑥 1 ) 𝑥 2 2 d .

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

Q4:

Use partial fractions to evaluate 𝑥 ( 𝑥 1 ) ( 𝑥 + 2 𝑥 + 1 ) 𝑥 2 2 d .

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

Q5:

Use partial fractions to find an analytic expression for the integral

  • A 2 ( 𝑥 ) + ( 𝑥 2 ) 1 𝑥 2 1 l n l n
  • B 2 ( 𝑥 ) + ( 2 𝑥 ) 1 𝑥 2 l n l n
  • C 2 ( 𝑥 ) + ( 2 𝑥 ) + 1 𝑥 2 1 l n l n
  • D 2 ( 𝑥 ) + ( 2 𝑥 ) 1 𝑥 2 1 l n l n
  • E 2 ( 𝑥 ) + ( 𝑥 2 ) + 1 𝑥 2 1 l n l n

Q6:

Find 𝐹 so that 𝐹 ( 𝑥 ) = 𝑥 ( 𝑥 𝑒 ) 2 and 𝐹 ( 0 ) = 2 .

  • A 𝐹 ( 𝑥 ) = | ( 𝑥 𝑒 ) | + 𝑒 𝑒 𝑥 l n
  • B 𝐹 ( 𝑥 ) = | ( 𝑒 𝑥 ) | 𝑥 𝑒 𝑥 l n
  • C 𝐹 ( 𝑥 ) = | ( 𝑥 𝑒 ) | 𝑥 𝑒 𝑥 l n
  • D 𝐹 ( 𝑥 ) = | ( 𝑒 𝑥 ) | + 𝑒 𝑒 𝑥 l n
  • E 𝐹 ( 𝑥 ) = | ( 𝑒 𝑥 ) | + 𝑒 𝑥 𝑒 l n

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