Worksheet: Integration by Parts

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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.

  • A 2 ๐‘ฅ ๐‘’ + ๏Šจ ๏— ๏Šฐ ๏Šจ C
  • B 2 ๏€ผ 1 2 ๐‘ฅ โˆ’ ๐‘ฅ + 1 ๏ˆ ๐‘’ + ๏Šจ ๏— ๏Šฐ ๏Šจ C
  • C 4 ๏€ผ 1 2 ๐‘ฅ โˆ’ ๐‘ฅ + 1 ๏ˆ ๐‘’ + ๏Šจ ๏— ๏Šฐ ๏Šจ C
  • D 4 ๏€น ๐‘ฅ โˆ’ ๐‘ฅ + 1 ๏… ๐‘’ + ๏Šจ ๏— ๏Šฐ ๏Šจ C
  • E 4 ๏€ผ 1 2 ๐‘ฅ โˆ’ ๐‘ฅ โˆ’ 1 ๏ˆ ๐‘’ + ๏Šจ ๏— ๏Šฐ ๏Šจ C

Q2:

Determine ๏„ธ9๐‘ฅ+7๐‘’๐‘ฅ๏Šซ๏—d.

  • A โˆ’ 9 5 ๏€ผ ๐‘ฅ + 4 4 4 5 ๏ˆ ๐‘’ + ๏Šฑ ๏Šซ ๏— C
  • B 9 5 ๏€ผ ๐‘ฅ + 4 4 4 5 ๏ˆ ๐‘’ + ๏Šฑ ๏Šซ ๏— C
  • C โˆ’ 1 5 ( 9 ๐‘ฅ + 7 ) ๐‘’ + ๏Šฑ ๏Šซ ๏— C
  • D โˆ’ 9 5 ๏€ผ ๐‘ฅ + 2 6 4 5 ๏ˆ ๐‘’ + ๏Šฑ ๏Šซ ๏— C
  • E โˆ’ 9 ๏€ผ ๐‘ฅ + 4 4 4 5 ๏ˆ ๐‘’ + ๏Šฑ ๏Šซ ๏— C

Q3:

Determine ๏„ธ(3๐‘ฅโˆ’5)๐‘ฅlnd.

  • A 1 3 ( 3 ๐‘ฅ โˆ’ 5 ) [ ( 3 ๐‘ฅ โˆ’ 5 ) + 1 ] + l n C
  • B 1 3 ( 3 ๐‘ฅ โˆ’ 5 ) ( 3 ๐‘ฅ โˆ’ 5 ) โˆ’ 1 + l n C
  • C 1 3 ( 3 ๐‘ฅ โˆ’ 5 ) [ ( 3 ๐‘ฅ โˆ’ 5 ) โˆ’ 1 ] + l n C
  • D 1 3 ( 3 ๐‘ฅ โˆ’ 5 ) ( 3 ๐‘ฅ โˆ’ 5 ) โˆ’ ๐‘ฅ + l n C

Q4:

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

  • A 4 3 โˆš ๐‘ฅ [ 8 ๐‘ฅ + 2 ] + l n C
  • B 2 3 โˆš ๐‘ฅ [ 8 ๐‘ฅ โˆ’ 2 ] + l n C
  • C 4 3 โˆš ๐‘ฅ 8 ๐‘ฅ โˆ’ 2 + l n C
  • D 4 3 โˆš ๐‘ฅ [ 8 ๐‘ฅ โˆ’ 2 ] + l n C

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?

  • A โˆ’ ๐‘ฅ ( 3 ๐‘ฅ + 7 ) 9 ๐‘ฅ l n
  • B โˆ’ ๐‘ฅ 2 ( 3 ๐‘ฅ + 1 4 ) + C
  • C ( โˆ’ 6 ๐‘ฅ โˆ’ 7 ) 9 ๐‘ฅ l n
  • D โˆ’ ๐‘ฅ ( 3 ๐‘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?

  • A ๐‘ฆ = 8 1 5 ( 2 ๐‘ฅ + 1 ) ( 3 ๐‘ฅ โˆ’ 1 ) + 1 ๏Žข ๏Žก
  • B ๐‘ฆ = 8 1 5 ( 2 ๐‘ฅ + 1 ) ( 3 ๐‘ฅ โˆ’ 1 ) โˆ’ 1 1 5 ๏Žข ๏Žก
  • C ๐‘ฆ = 4 1 5 ( 2 ๐‘ฅ + 1 ) ( 8 ๐‘ฅ โˆ’ 1 ) + 1 1 1 5 ๏Žข ๏Žก
  • D ๐‘ฆ = 8 1 5 ( 2 ๐‘ฅ + 1 ) ( 3 ๐‘ฅ โˆ’ 1 ) โˆ’ 1 6 1 5 ๏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.

  • A ๐‘“ ( ๐‘ฅ ) = โˆ’ 3 ๐‘’ 4 ( 2 ๐‘ฅ + 1 ) + 2 1 4 ๐‘’ ๏Šจ ๏— ๏Šจ
  • B ๐‘“ ( ๐‘ฅ ) = ๐‘’ 4 ( 2 ๐‘ฅ + 1 ) + 1 1 1 2 ๐‘’ ๏Šจ ๏— ๏Šจ
  • C ๐‘“ ( ๐‘ฅ ) = 3 ๐‘’ 4 ( 2 ๐‘ฅ + 1 ) + 1 9 4 ๐‘’ ๏Šจ ๏— ๏Šจ
  • D ๐‘“ ( ๐‘ฅ ) = 3 ๐‘ฅ ๐‘’ 4 ( 2 ๐‘ฅ + 1 ) + 1 9 4 ๐‘’ ๏Šจ ๏— ๏Šจ

Q8:

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

  • A ๐‘ฅ ๐‘ฅ โˆ’ ๐‘ฅ + c o s s i n C
  • B ๐‘ฅ ( ๐‘ฅ โˆ’ ๐‘ฅ ) + s i n c o s C
  • C s i n c o s C ๐‘ฅ + ๐‘ฅ ๐‘ฅ +
  • D โˆ’ ๐‘ฅ โˆ’ ๐‘ฅ ๐‘ฅ + s i n c o s C
  • E s i n c o s C ๐‘ฅ โˆ’ ๐‘ฅ ๐‘ฅ +

Q9:

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

  • A 2 ๐‘’ ( ๐‘ฅ + ๐‘ฅ ) + ๏— s i n c o s C
  • B 1 2 ๐‘’ ( ๐‘ฅ โˆ’ ๐‘ฅ ) + ๏— s i n c o s C
  • C ๐‘’ ( ๐‘ฅ + ๐‘ฅ ) + ๏— s i n c o s C
  • D 2 ๐‘’ ( ๐‘ฅ โˆ’ ๐‘ฅ ) + ๏— s i n c o s C
  • E 1 2 ๐‘’ ( ๐‘ฅ + ๐‘ฅ ) + ๏— s i n c o s C

Q10:

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

  • A l n C ๐‘ฅ โˆ’ ๐‘ฅ +
  • B ๐‘ฅ ๐‘ฅ + 1 + l n C
  • C ๐‘ฅ ( ๐‘ฅ + 1 ) + l n C
  • D ๐‘ฅ ( ๐‘ฅ โˆ’ 1 ) + l n C
  • E ๐‘ฅ ๐‘ฅ โˆ’ 1 + l n C

Q11:

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

  • A ( โˆ’ 5 ๐‘ฅ + 1 2 ) ๐‘ฅ โˆ’ 5 ๐‘ฅ + c o s s i n C
  • B ( 5 ๐‘ฅ โˆ’ 1 2 ) ๐‘ฅ โˆ’ 5 ๐‘ฅ + c o s s i n C
  • C ( 5 ๐‘ฅ โˆ’ 1 2 ) ๐‘ฅ + 5 ๐‘ฅ + c o s s i n C
  • D ( โˆ’ 5 ๐‘ฅ + 1 2 ) ๐‘ฅ + 5 ๐‘ฅ + c o s s i n C

Q12:

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

  • A ๐‘’ ๏€ผ 9 2 ๐‘ฅ + 6 ๐‘ฅ + 1 0 ๏ˆ + ๏— ๏Šจ C
  • B ๐‘’ ๏€ผ 9 2 ๐‘ฅ + 3 ๐‘ฅ + 1 ๏ˆ + ๏— ๏Šจ C
  • C ๐‘’ ๏€น 9 ๐‘ฅ + 3 ๐‘ฅ + 1 0 ๏… + ๏— ๏Šจ C
  • D ๐‘’ ๏€น 9 ๐‘ฅ + 6 ๐‘ฅ + 1 0 ๏… + ๏— ๏Šจ C

Q13:

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

  • A 2 ๐‘’ 3 ( ๐‘ฅ + 1 ) + ๏— C
  • B 2 ๐‘’ ( 2 ๐‘ฅ + 1 ) 3 ( ๐‘ฅ + 1 ) + ๏— C
  • C โˆ’ 2 ๐‘’ ( 2 ๐‘ฅ + 1 ) 3 ( ๐‘ฅ + 1 ) + ๏— C
  • D โˆ’ 2 ๐‘’ 3 ( ๐‘ฅ + 1 ) + ๏— C

Q14:

Determine ๏„ธ64๐‘ฅ๐‘ฅlnd๏Šซ.

  • A 6 ๐‘ฅ ๏€น 4 ๐‘ฅ โˆ’ 5 ๏… + l n C ๏Šซ
  • B ๐‘ฅ ๏€น 4 ๐‘ฅ โˆ’ 5 ๏… + l n C ๏Šซ
  • C 6 ๏€น 4 ๐‘ฅ โˆ’ 5 ๏… + l n C ๏Šซ
  • D 6 ๐‘ฅ ๏€น 4 ๐‘ฅ + 5 ๏… + l n C ๏Šซ
  • E 6 ๐‘ฅ ๏€น 4 ๐‘ฅ โˆ’ 5 ๏… + ๏Šจ ๏Šซ l n C

Q15:

Determine ๏„ธ๐‘ฅ(5๐‘ฅ)๐‘ฅlnd๏Šจ.

  • A 1 4 ๐‘ฅ ๏€บ 2 ( 5 ๐‘ฅ ) + 2 5 ๐‘ฅ โˆ’ 1 ๏† + ๏Šจ ๏Šจ l n l n C
  • B 1 4 ๐‘ฅ ๏€บ 2 ( 5 ๐‘ฅ ) โˆ’ 2 5 ๐‘ฅ + 1 ๏† + ๏Šจ ๏Šจ l n l n C
  • C 1 4 ๐‘ฅ ๏€บ 2 ( 5 ๐‘ฅ ) โˆ’ 2 5 ๐‘ฅ + 1 ๏† + l n l n C ๏Šจ
  • D 1 2 ๐‘ฅ ๏€บ 2 ( 5 ๐‘ฅ ) โˆ’ 2 5 ๐‘ฅ + 1 ๏† + ๏Šจ ๏Šจ l n l n C
  • E 1 2 ๐‘ฅ ( 5 ๐‘ฅ ) + ๏Šจ ๏Šจ l n C

Q16:

Determine ๏„ธ93๐‘ฅ๐‘ฅ๐‘ฅlnd๏Šฌ.

  • A 9 2 5 ๐‘ฅ ( โˆ’ 5 3 ๐‘ฅ โˆ’ 1 ) + ๏Šฑ ๏Šซ l n C
  • B 9 2 5 ๐‘ฅ ( โˆ’ 5 3 ๐‘ฅ + 1 ) + ๏Šฑ ๏Šซ l n C
  • C โˆ’ 9 5 ๐‘ฅ 3 ๐‘ฅ + ๏Šฑ ๏Šซ l n C
  • D 9 2 5 ๐‘ฅ ( 3 ๐‘ฅ โˆ’ 1 ) + ๏Šฑ ๏Šซ l n C
  • E 9 5 ๐‘ฅ ( โˆ’ 5 3 ๐‘ฅ โˆ’ 1 ) + ๏Šฑ ๏Šซ l n C

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.

  • A ๐‘“ ( ๐‘ฅ ) = โˆ’ 7 ๐‘’ 4 ( 2 ๐‘ฅ + 1 ) + 1 0 3 1 2 ๐‘’ ๏Šจ ๏— ๏Šจ
  • B ๐‘“ ( ๐‘ฅ ) = ๐‘’ 4 ( 2 ๐‘ฅ + 1 ) + 1 1 1 2 ๐‘’ ๏Šจ ๏— ๏Šจ
  • C ๐‘“ ( ๐‘ฅ ) = 7 ๐‘’ 4 ( 2 ๐‘ฅ + 1 ) + 8 9 1 2 ๐‘’ ๏Šจ ๏— ๏Šจ
  • D ๐‘“ ( ๐‘ฅ ) = 7 ๐‘ฅ ๐‘’ 4 ( 2 ๐‘ฅ + 1 ) + 8 9 1 2 ๐‘’ ๏Šจ ๏— ๏Šจ

Q18:

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

  • A 1 4 โˆ’ ๐œ‹ 8
  • B ๐œ‹ 8 โˆ’ 1 4
  • C ๐œ‹ 8 + 1 4
  • D 1 4 โˆ’ ๐œ‹ 4
  • E ๐œ‹ 4 โˆ’ 1 4

Q19:

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

  • A ๐‘’ โˆ’ 2
  • B 1 โˆ’ ๐‘’
  • C ๐‘’ + 2
  • D 2 โˆ’ 3 ๐‘’
  • E ๐‘’ โˆ’ 1

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