Worksheet: Integration by Parts

In this worksheet, we will practice using integration by parts to find the integral of a product of functions.

Q1:

Determine ๏„ธ2๐‘ฅ๐‘’๐‘ฅ๏Šจ๏—๏Šฐ๏Šจd.

  • A2๐‘ฅ๐‘’+๏Šจ๏—๏Šฐ๏ŠจC
  • B2๏€ผ12๐‘ฅโˆ’๐‘ฅ+1๏ˆ๐‘’+๏Šจ๏—๏Šฐ๏ŠจC
  • C4๏€ผ12๐‘ฅโˆ’๐‘ฅ+1๏ˆ๐‘’+๏Šจ๏—๏Šฐ๏ŠจC
  • D4๏€น๐‘ฅโˆ’๐‘ฅ+1๏…๐‘’+๏Šจ๏—๏Šฐ๏ŠจC
  • E4๏€ผ12๐‘ฅโˆ’๐‘ฅโˆ’1๏ˆ๐‘’+๏Šจ๏—๏Šฐ๏ŠจC

Q2:

Determine ๏„ธ9๐‘ฅ+7๐‘’๐‘ฅ๏Šซ๏—d.

  • Aโˆ’95๏€ผ๐‘ฅ+4445๏ˆ๐‘’+๏Šฑ๏Šซ๏—C
  • B95๏€ผ๐‘ฅ+4445๏ˆ๐‘’+๏Šฑ๏Šซ๏—C
  • Cโˆ’15(9๐‘ฅ+7)๐‘’+๏Šฑ๏Šซ๏—C
  • Dโˆ’95๏€ผ๐‘ฅ+2645๏ˆ๐‘’+๏Šฑ๏Šซ๏—C
  • Eโˆ’9๏€ผ๐‘ฅ+4445๏ˆ๐‘’+๏Šฑ๏Šซ๏—C

Q3:

Determine ๏„ธ(3๐‘ฅโˆ’5)๐‘ฅlnd.

  • A13(3๐‘ฅโˆ’5)[(3๐‘ฅโˆ’5)+1]+lnC
  • B13(3๐‘ฅโˆ’5)(3๐‘ฅโˆ’5)โˆ’1+lnC
  • C13(3๐‘ฅโˆ’5)[(3๐‘ฅโˆ’5)โˆ’1]+lnC
  • D13(3๐‘ฅโˆ’5)(3๐‘ฅโˆ’5)โˆ’๐‘ฅ+lnC

Q4:

Determine ๏„ธ๏€บ28๐‘ฅรท3โˆš๐‘ฅ๏†๐‘ฅlnd.

  • A43โˆš๐‘ฅ[8๐‘ฅ+2]+lnC
  • B23โˆš๐‘ฅ[8๐‘ฅโˆ’2]+lnC
  • C43โˆš๐‘ฅ8๐‘ฅโˆ’2+lnC
  • D43โˆš๐‘ฅ[8๐‘ฅโˆ’2]+lnC

Q5:

Suppose ๏„ธ(โˆ’6๐‘ฅโˆ’7)9๐‘ฅ๐‘ฅ=๐‘ฆ๐‘งโˆ’๏„ธ๐‘ง๐‘ฆlndd. Which of the following is equal to ๐‘ฆ๐‘ง?

  • Aโˆ’๐‘ฅ(3๐‘ฅ+7)9๐‘ฅln
  • Bโˆ’๐‘ฅ2(3๐‘ฅ+14)+C
  • C(โˆ’6๐‘ฅโˆ’7)9๐‘ฅln
  • Dโˆ’๐‘ฅ(3๐‘ฅ+7)

Q6:

A curve passes through ๏€ผ0,715๏ˆ and the tangent at its point (๐‘ฅ,๐‘ฆ) has slope 8๐‘ฅโˆš2๐‘ฅ+1. What is the equation of the curve?

  • A๐‘ฆ=815(2๐‘ฅ+1)(3๐‘ฅโˆ’1)+1๏Žข๏Žก
  • B๐‘ฆ=815(2๐‘ฅ+1)(3๐‘ฅโˆ’1)โˆ’115๏Žข๏Žก
  • C๐‘ฆ=415(2๐‘ฅ+1)(8๐‘ฅโˆ’1)+1115๏Žข๏Žก
  • D๐‘ฆ=815(2๐‘ฅ+1)(3๐‘ฅโˆ’1)โˆ’1615๏Žข๏Žก

Q7:

The slope of the tangent to the curve ๐‘ฆ=๐‘“(๐‘ฅ) at the point (๐‘ฅ,๐‘ฆ) is given by 3๐‘ฅ๐‘’(2๐‘ฅ+1)๏Šจ๏—๏Šจ. Determine ๐‘“(๐‘ฅ) if the point ๏€น1,5๐‘’๏…๏Šจ lies on the curve.

  • A๐‘“(๐‘ฅ)=โˆ’3๐‘’4(2๐‘ฅ+1)+214๐‘’๏Šจ๏—๏Šจ
  • B๐‘“(๐‘ฅ)=๐‘’4(2๐‘ฅ+1)+1112๐‘’๏Šจ๏—๏Šจ
  • C๐‘“(๐‘ฅ)=3๐‘’4(2๐‘ฅ+1)+194๐‘’๏Šจ๏—๏Šจ
  • D๐‘“(๐‘ฅ)=3๐‘ฅ๐‘’4(2๐‘ฅ+1)+194๐‘’๏Šจ๏—๏Šจ

Q8:

Use integration by parts to evaluate ๏„ธ๐‘ฅ๐‘ฅ๐‘ฅsind.

  • A๐‘ฅ๐‘ฅโˆ’๐‘ฅ+cossinC
  • B๐‘ฅ(๐‘ฅโˆ’๐‘ฅ)+sincosC
  • CsincosC๐‘ฅ+๐‘ฅ๐‘ฅ+
  • Dโˆ’๐‘ฅโˆ’๐‘ฅ๐‘ฅ+sincosC
  • EsincosC๐‘ฅโˆ’๐‘ฅ๐‘ฅ+

Q9:

By setting ๐‘ข=๐‘’๏— and dcosd๐‘ฃ=๐‘ฅ๐‘ฅ, evaluate ๏„ธ๐‘’๐‘ฅ๐‘ฅ๏—cosd by integrating by parts.

  • A2๐‘’(๐‘ฅ+๐‘ฅ)+๏—sincosC
  • B12๐‘’(๐‘ฅโˆ’๐‘ฅ)+๏—sincosC
  • C๐‘’(๐‘ฅ+๐‘ฅ)+๏—sincosC
  • D2๐‘’(๐‘ฅโˆ’๐‘ฅ)+๏—sincosC
  • E12๐‘’(๐‘ฅ+๐‘ฅ)+๏—sincosC

Q10:

Integrate ๏„ธ๐‘ฅ๐‘ฅlnd by parts using ๐‘ข=๐‘ฅln and dd๐‘ฃ=๐‘ฅ.

  • AlnC๐‘ฅโˆ’๐‘ฅ+
  • B๐‘ฅ๐‘ฅ+1+lnC
  • C๐‘ฅ(๐‘ฅ+1)+lnC
  • D๐‘ฅ(๐‘ฅโˆ’1)+lnC
  • E๐‘ฅ๐‘ฅโˆ’1+lnC

Q11:

Determine ๏„ธ(5๐‘ฅโˆ’12)๐‘ฅ๐‘ฅsind.

  • A(โˆ’5๐‘ฅ+12)๐‘ฅโˆ’5๐‘ฅ+cossinC
  • B(5๐‘ฅโˆ’12)๐‘ฅโˆ’5๐‘ฅ+cossinC
  • C(5๐‘ฅโˆ’12)๐‘ฅ+5๐‘ฅ+cossinC
  • D(โˆ’5๐‘ฅ+12)๐‘ฅ+5๐‘ฅ+cossinC

Q12:

Determine ๏„ธ(3๐‘ฅ+4)๐‘’๐‘ฅ๏Šจ๏—d.

  • A๐‘’๏€ผ92๐‘ฅ+6๐‘ฅ+10๏ˆ+๏—๏ŠจC
  • B๐‘’๏€ผ92๐‘ฅ+3๐‘ฅ+1๏ˆ+๏—๏ŠจC
  • C๐‘’๏€น9๐‘ฅ+3๐‘ฅ+10๏…+๏—๏ŠจC
  • D๐‘’๏€น9๐‘ฅ+6๐‘ฅ+10๏…+๏—๏ŠจC

Q13:

Determine ๏„ธ2๐‘’๐‘ฅ3(๐‘ฅ+1)๐‘ฅ๏—๏Šจd.

  • A2๐‘’3(๐‘ฅ+1)+๏—C
  • B2๐‘’(2๐‘ฅ+1)3(๐‘ฅ+1)+๏—C
  • Cโˆ’2๐‘’(2๐‘ฅ+1)3(๐‘ฅ+1)+๏—C
  • Dโˆ’2๐‘’3(๐‘ฅ+1)+๏—C

Q14:

Determine ๏„ธ64๐‘ฅ๐‘ฅlnd๏Šซ.

  • A6๐‘ฅ๏€น4๐‘ฅ+5๏…+lnC๏Šซ
  • B6๏€น4๐‘ฅโˆ’5๏…+lnC๏Šซ
  • C6๐‘ฅ๏€น4๐‘ฅโˆ’5๏…+lnC๏Šซ
  • D6๐‘ฅ๏€น4๐‘ฅโˆ’5๏…+๏Šจ๏ŠซlnC
  • E๐‘ฅ๏€น4๐‘ฅโˆ’5๏…+lnC๏Šซ

Q15:

Determine ๏„ธ๐‘ฅ(5๐‘ฅ)๐‘ฅlnd๏Šจ.

  • A14๐‘ฅ๏€บ2(5๐‘ฅ)+25๐‘ฅโˆ’1๏†+๏Šจ๏ŠจlnlnC
  • B14๐‘ฅ๏€บ2(5๐‘ฅ)โˆ’25๐‘ฅ+1๏†+๏Šจ๏ŠจlnlnC
  • C14๐‘ฅ๏€บ2(5๐‘ฅ)โˆ’25๐‘ฅ+1๏†+lnlnC๏Šจ
  • D12๐‘ฅ๏€บ2(5๐‘ฅ)โˆ’25๐‘ฅ+1๏†+๏Šจ๏ŠจlnlnC
  • E12๐‘ฅ(5๐‘ฅ)+๏Šจ๏ŠจlnC

Q16:

Determine ๏„ธ93๐‘ฅ๐‘ฅ๐‘ฅlnd๏Šฌ.

  • A925๐‘ฅ(โˆ’53๐‘ฅโˆ’1)+๏Šฑ๏ŠซlnC
  • B925๐‘ฅ(โˆ’53๐‘ฅ+1)+๏Šฑ๏ŠซlnC
  • Cโˆ’95๐‘ฅ3๐‘ฅ+๏Šฑ๏ŠซlnC
  • D925๐‘ฅ(3๐‘ฅโˆ’1)+๏Šฑ๏ŠซlnC
  • E95๐‘ฅ(โˆ’53๐‘ฅโˆ’1)+๏Šฑ๏ŠซlnC

Q17:

The slope of the tangent to the curve ๐‘ฆ=๐‘“(๐‘ฅ) at the point (๐‘ฅ,๐‘ฆ) is given by 7๐‘ฅ๐‘’(2๐‘ฅ+1)๏Šจ๏—๏Šจ. Determine ๐‘“(๐‘ฅ) if the point ๏€น1,8๐‘’๏…๏Šจ lies on the curve.

  • A๐‘“(๐‘ฅ)=โˆ’7๐‘’4(2๐‘ฅ+1)+10312๐‘’๏Šจ๏—๏Šจ
  • B๐‘“(๐‘ฅ)=๐‘’4(2๐‘ฅ+1)+1112๐‘’๏Šจ๏—๏Šจ
  • C๐‘“(๐‘ฅ)=7๐‘’4(2๐‘ฅ+1)+8912๐‘’๏Šจ๏—๏Šจ
  • D๐‘“(๐‘ฅ)=7๐‘ฅ๐‘’4(2๐‘ฅ+1)+8912๐‘’๏Šจ๏—๏Šจ

Q18:

Use integration by parts to find the exact value of ๏„ธ๐‘ฅ2๐‘ฅ๐‘ฅ๏Ž„/๏Šช๏Šฆ๏Šจsind.

  • A14โˆ’๐œ‹8
  • B๐œ‹8โˆ’14
  • C๐œ‹8+14
  • D14โˆ’๐œ‹4
  • E๐œ‹4โˆ’14

Q19:

Evaluate ๏„ธ๐‘ฅ๐‘’๐‘ฅ๏Šง๏Šฆ๏Šจ๏—d.

  • A๐‘’โˆ’2
  • B1โˆ’๐‘’
  • C๐‘’+2
  • D2โˆ’3๐‘’
  • E๐‘’โˆ’1

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