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=𝑥𝑒 for 𝑛=0,1,2,.

What is 𝐼?

  • A𝑥𝑒+C
  • B𝑥𝑒+C
  • C𝑒+C
  • D𝑒+C
  • E𝑥+C

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

  • A(𝑛+1)𝐼=𝑥𝑒+𝐼
  • B(𝑛1)𝐼=𝑥𝑒+𝐼
  • C(𝑛+1)𝐼=𝑥𝑒𝐼
  • D(𝑛1)𝐼=𝑥𝑒𝐼
  • E𝐼=𝑥𝑒𝐼

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

  • A𝐼=𝑥𝑒+𝑛𝐼
  • B𝐼𝑛=𝑥𝑒+𝐼
  • C𝑛𝐼=𝑥𝑒𝐼
  • D𝐼=𝑥𝑒𝑛𝐼
  • E𝐼𝑛=𝑥𝑒𝐼

Evaluate the indefinite integral 𝑥𝑒𝑥d.

  • A𝑥𝑒3𝑥𝑒6𝑥𝑒6𝑒+C
  • B𝑥𝑒3𝑥𝑒2𝑥𝑒2𝑒+C
  • C𝑥𝑒3𝑥𝑒+2𝑥𝑒𝑒+C
  • D𝑥𝑒3𝑥𝑒+6𝑥𝑒+6𝑒+C
  • E𝑥𝑒3𝑥𝑒+6𝑥𝑒6𝑒+C

Q2:

The following table lists the integrals 𝐼=𝑥𝑥𝑥lnd for 𝑛=0,1, and 2.

𝑛012
𝐼𝑥(𝑥1)+𝐾ln14𝑥(2𝑥1)+𝐾ln19𝑥(3𝑥1)+𝐾ln

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

  • A𝐼=1(𝑛+1)𝑥((𝑛+1)𝑥1)+𝐾ln
  • B𝐼=1(𝑛1)𝑥((𝑛1)𝑥1)+𝐾ln
  • C𝐼=1(𝑛+1)𝑥((𝑛+1)𝑥1)+𝐾ln
  • D𝐼=1𝑛𝑥(𝑛𝑥1)+𝐾ln
  • E𝐼=1(𝑛+1)𝑥(𝑛𝑥1)+𝐾ln

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

  • A𝑢=𝑥ln, dd𝑣=𝑥𝑥
  • B𝑢=𝑥, dlnd𝑣=𝑥𝑥

Q3:

Evaluate the indefinite integral 𝑥(12+𝑥)d.

  • A𝑥24(12+𝑥)+112(𝑥)+arctanc
  • B𝑥24(12+𝑥)+1483𝑥23+arctanc
  • C𝑥24(12+𝑥)+1483𝑥24+arctanc
  • D𝑥24(12+𝑥)+124𝑥23+arctanc
  • E𝑥12(12+𝑥)+1483𝑥23+arctanc

Q4:

Evaluate the indefinite integral (𝑥)sin.

  • A132(𝑥)+14(2𝑥)+38𝑥+sinsinc
  • B132(𝑥)14(2𝑥)+38𝑥+sinsinc
  • C132(4𝑥)+14(2𝑥)+38𝑥+sinsinc
  • D132(4𝑥)14(2𝑥)+38𝑥+sinsinc
  • E132(4𝑥)14(𝑥)+38𝑥+sinsinc

Q5:

Evaluate the indefinite integral 𝑥𝑒𝑥d.

  • A12𝑥𝑒+2𝑥𝑒+2𝑥𝑒+2𝑒+c
  • B12𝑥𝑒+34𝑥𝑒+34𝑥𝑒+38𝑒+c
  • C12𝑥𝑒2𝑥𝑒+2𝑥𝑒2𝑒+c
  • D12𝑥𝑒2𝑥𝑒+2𝑥𝑒2𝑥𝑒+c
  • E12𝑥𝑒34𝑥𝑒+34𝑥𝑒38𝑒+c

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