Worksheet: Reduction Formulae for Integration

In this worksheet, we will practice deriving reduction formulae and using them to evaluate integrals.

Q1:

Reduction formulas relate integrals involving an integer parameter. Let 𝐼 = 𝑥 𝑒 for 𝑛 = 0 , 1 , 2 , .

What is 𝐼 ?

  • A 𝑒 + C
  • B 𝑥 𝑒 + C
  • C 𝑥 𝑒 + C
  • D 𝑥 + C
  • E 𝑒 + C

Use 𝑢 = 𝑒 , d d 𝑣 = 𝑥 𝑥 and integration by parts to derive a relation between 𝐼 and 𝐼 . Clear denominators if necessary.

  • A ( 𝑛 + 1 ) 𝐼 = 𝑥 𝑒 𝐼
  • B ( 𝑛 1 ) 𝐼 = 𝑥 𝑒 + 𝐼
  • C 𝐼 = 𝑥 𝑒 𝐼
  • D ( 𝑛 1 ) 𝐼 = 𝑥 𝑒 𝐼
  • E ( 𝑛 + 1 ) 𝐼 = 𝑥 𝑒 + 𝐼

From your previous answer, find a formula for 𝐼 in terms of 𝐼 .

  • A 𝐼 𝑛 = 𝑥 𝑒 𝐼
  • B 𝑛 𝐼 = 𝑥 𝑒 𝐼
  • C 𝐼 𝑛 = 𝑥 𝑒 + 𝐼
  • D 𝐼 = 𝑥 𝑒 + 𝑛 𝐼
  • E 𝐼 = 𝑥 𝑒 𝑛 𝐼

Evaluate the indefinite integral 𝑥 𝑒 𝑥 d .

  • A 𝑥 𝑒 3 𝑥 𝑒 + 6 𝑥 𝑒 6 𝑒 + C
  • B 𝑥 𝑒 3 𝑥 𝑒 + 6 𝑥 𝑒 + 6 𝑒 + C
  • C 𝑥 𝑒 3 𝑥 𝑒 6 𝑥 𝑒 6 𝑒 + C
  • D 𝑥 𝑒 3 𝑥 𝑒 + 2 𝑥 𝑒 𝑒 + C
  • E 𝑥 𝑒 3 𝑥 𝑒 2 𝑥 𝑒 2 𝑒 + C

Q2:

The following table lists the integrals for , and 2.

Use the table to predict a formula for and then verify this by differentiating.

  • A
  • B
  • C
  • D
  • E

Write the and with which you can prove the formula using integration by parts.

  • A ,
  • B ,

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