Worksheet: Integration by Substitution: Definite Integrals

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In this worksheet, we will practice using integration by substitution for definite integrals.

Q1:

Use an appropriate substitution followed by a trigonometric one to evaluate 𝑦 𝑦 √ 1 + ( 𝑦 ) 𝑒 1 2 d l n .

  • A l n 2 − √ 2
  • B l n 2 + √ 2
  • C l n ( 2 )
  • D l n 1 + √ 2
  • E l n √ 2

Q2:

Evaluate 2 𝑡 𝑡 𝑡 c s c c o t d .

  • A − 2 √ 3
  • B 2 √ 3
  • C − 2
  • D2
  • E0

Q3:

Find the average value of on the interval .

  • A
  • B
  • C
  • D
  • E

Q4:

Find the average value of 𝑓 ( 𝑡 ) = 𝑒 𝑡 s i n 𝑡 c o s on the interval 0 , 𝜋 2 .

  • A − 1 + 𝑒
  • B − 2 𝑒 𝜋 + 2 𝜋
  • C − 𝜋 2 + 𝑒 𝜋 2
  • D − 2 𝜋 + 2 𝑒 𝜋
  • E − 1

Q5:

Find the average value of on the interval .

  • A
  • B
  • C
  • D

Q6:

Find 2 𝑥 9 + 2 𝑥 d 𝑥 to the nearest hundredth.

Q7:

Evaluate ( 2 − 𝑟 ) 𝑟 d .

  • A 1 1 3
  • B81
  • C − 8 1
  • D 1 4
  • E − 1 4

Q8:

Determine 8 𝑥 − 8 √ 𝑥 − 1 𝑥 d to the nearest hundredth.

Q9:

Evaluate 𝑥 √ 5 𝑥 + 3 d 𝑥 to the nearest thousandth.

Q10:

Find to the nearest thousandth.

Q11:

Determine 4 𝑥 ( 𝑥 − 6 ) 𝑥 d rounded to one decimal place.

Q12:

Find − 9 ( 𝑧 ) ( 𝑧 ) 𝑧 t a n s e c d .

  • A 9 2
  • B 1 2
  • C − 9
  • D − 9 2

Q13:

Use an appropriate substitution followed by a trigonometric one to evaluate 𝑒 𝑡 √ 𝑒 + 9 l n 4 0 𝑡 2 𝑡 d .

  • A l n l n 9 + 1 + √ 1 0
  • B l n l n 1 0 − 1 + √ 1 0
  • C l n l n 3 − 1 + √ 1 0
  • D l n l n 9 − 1 + √ 1 0
  • E l n l n 3 + 1 − √ 1 0

Q14:

Find the average value of on the interval .

  • A
  • B
  • C
  • D
  • E

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