Worksheet: Integration by Parts for Indefinite Integrals

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

Q1:

Determine ๏„ธ 2 ๐‘ฅ ๐‘’ ๐‘ฅ ๏Šจ ๏— ๏Šฐ ๏Šจ d .

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

Q2:

Determine ๏„ธ 9 ๐‘ฅ + 7 ๐‘’ ๐‘ฅ 5 ๐‘ฅ d .

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

Q3:

Determine ๏„ธ ( 3 ๐‘ฅ โˆ’ 5 ) ๐‘ฅ l n d .

  • 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’ ๐‘ฅ + 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’ 1 ] + l n C

Q4:

Determine ๏„ธ ๏€บ 2 8 ๐‘ฅ รท 3 โˆš ๐‘ฅ ๏† ๐‘ฅ l n d .

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

Q5:

Suppose ๏„ธ ( โˆ’ 6 ๐‘ฅ โˆ’ 7 ) 9 ๐‘ฅ ๐‘ฅ = ๐‘ฆ ๐‘ง โˆ’ ๏„ธ ๐‘ง ๐‘ฆ l n d d . Which of the following is equal to ๐‘ฆ ๐‘ง ?

  • A โˆ’ ๐‘ฅ ( 3 ๐‘ฅ + 7 )
  • 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 ) 9 ๐‘ฅ l n

Q6:

A curve passes through ๏€ผ 0 , 7 1 5 ๏ˆ and the tangent at its point ( ๐‘ฅ , ๐‘ฆ ) has slope 8 ๐‘ฅ โˆš 2 ๐‘ฅ + 1 . What is the equation of the curve?

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

Q7:

The slope of the tangent to the curve ๐‘ฆ = ๐‘“ ( ๐‘ฅ ) at the point ( ๐‘ฅ , ๐‘ฆ ) is given by 3 ๐‘ฅ ๐‘’ ( 2 ๐‘ฅ + 1 ) 2 ๐‘ฅ 2 . Determine ๐‘“ ( ๐‘ฅ ) if the point ๏€น 1 , 5 ๐‘’ ๏… 2 lies on the curve.

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

Q8:

Use integration by parts to evaluate ๏„ธ ๐‘ฅ ๐‘ฅ ๐‘ฅ s i n d .

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

Q9:

By setting ๐‘ข = ๐‘’ ๏— and d c o s d ๐‘ฃ = ๐‘ฅ ๐‘ฅ , evaluate ๏„ธ ๐‘’ ๐‘ฅ ๐‘ฅ ๏— c o s d by integrating by parts.

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

Q10:

Integrate ๏„ธ ๐‘ฅ ๐‘ฅ l n d by parts using ๐‘ข = ๐‘ฅ l n and d d ๐‘ฃ = ๐‘ฅ .

  • A ๐‘ฅ ๐‘ฅ โˆ’ 1 + l n C
  • 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 l n C ๐‘ฅ โˆ’ ๐‘ฅ +

Q11:

Determine ๏„ธ ( 5 ๐‘ฅ โˆ’ 1 2 ) ๐‘ฅ ๐‘ฅ s i n d .

  • 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 + 5 ๐‘ฅ + c o s s i n C
  • C ( 5 ๐‘ฅ โˆ’ 1 2 ) ๐‘ฅ โˆ’ 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 ( 3 ๐‘ฅ + 4 ) ๐‘’ ๐‘ฅ 2 ๐‘ฅ d .

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

Q13:

Determine ๏„ธ 2 ๐‘’ ๐‘ฅ 3 ( ๐‘ฅ + 1 ) ๐‘ฅ ๐‘ฅ 2 d .

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

Q14:

Determine ๏„ธ 6 4 ๐‘ฅ ๐‘ฅ l n d 5 .

  • A ๐‘ฅ ๏€น 4 ๐‘ฅ โˆ’ 5 ๏… + l n C 5
  • B 6 ๐‘ฅ ๏€น 4 ๐‘ฅ + 5 ๏… + l n C 5
  • C 6 ๏€น 4 ๐‘ฅ โˆ’ 5 ๏… + l n C 5
  • D 6 ๐‘ฅ ๏€น 4 ๐‘ฅ โˆ’ 5 ๏… + l n C 5
  • E 6 ๐‘ฅ ๏€น 4 ๐‘ฅ โˆ’ 5 ๏… + 2 5 l n C

Q15:

Determine ๏„ธ ๐‘ฅ ( 5 ๐‘ฅ ) ๐‘ฅ l n d ๏Šจ .

  • A 1 2 ๐‘ฅ ๏€บ 2 ( 5 ๐‘ฅ ) โˆ’ 2 5 ๐‘ฅ + 1 ๏† + ๏Šจ ๏Šจ l n l n C
  • B 1 4 ๐‘ฅ ๏€บ 2 ( 5 ๐‘ฅ ) + 2 5 ๐‘ฅ โˆ’ 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 4 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 ๏„ธ 9 3 ๐‘ฅ ๐‘ฅ ๐‘ฅ l n d 6 .

  • A 9 5 ๐‘ฅ ( โˆ’ 5 3 ๐‘ฅ โˆ’ 1 ) + โˆ’ 5 l n C
  • B 9 2 5 ๐‘ฅ ( โˆ’ 5 3 ๐‘ฅ + 1 ) + โˆ’ 5 l n C
  • C 9 2 5 ๐‘ฅ ( 3 ๐‘ฅ โˆ’ 1 ) + โˆ’ 5 l n C
  • D 9 2 5 ๐‘ฅ ( โˆ’ 5 3 ๐‘ฅ โˆ’ 1 ) + โˆ’ 5 l n C
  • E โˆ’ 9 5 ๐‘ฅ 3 ๐‘ฅ + โˆ’ 5 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 7 ๐‘ฅ ๐‘’ ( 2 ๐‘ฅ + 1 ) 2 ๐‘ฅ 2 . Determine ๐‘“ ( ๐‘ฅ ) if the point ๏€น 1 , 8 ๐‘’ ๏… 2 lies on the curve.

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

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