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Worksheet: Evaluating Definite Integrals by Substitution

Q1:

Evaluate ο„Έ 2 𝑑 𝑑 𝑑 πœ‹ 2 πœ‹ 6 c s c c o t d .

  • A βˆ’ 2 √ 3
  • B 2 √ 3
  • C βˆ’ 2
  • D2
  • E0

Q2:

Find the average value of 𝑓 ( π‘₯ ) = π‘₯ π‘₯ s i n c o s 4 on the interval  0 , πœ‹ 2  .

  • A 1 5
  • B βˆ’ 2 5 πœ‹
  • C πœ‹ 1 0
  • D 2 5 πœ‹
  • E 2 πœ‹

Q3:

Find the average value of 𝑓 ( π‘₯ ) = π‘₯ π‘₯ s i n c o s 4 on the interval [ 0 , πœ‹ ] .

  • A 2 5
  • B βˆ’ 2 5 πœ‹
  • C 2 πœ‹ 5
  • D 2 5 πœ‹
  • E 2 πœ‹

Q4:

Find the average value of 𝑓 ( 𝑑 ) = 𝑒 𝑑 s i n 𝑑 c o s on the interval  0 , πœ‹ 2  .

  • A βˆ’ 1 + 𝑒
  • B βˆ’ 2 𝑒 πœ‹ + 2 πœ‹
  • C βˆ’ πœ‹ 2 + 𝑒 πœ‹ 2
  • D βˆ’ 2 πœ‹ + 2 𝑒 πœ‹
  • E βˆ’ 1

Q5:

Find the average value of 𝑔 ( 𝑑 ) = 2 𝑑 √ 2 𝑑 + 2 2 on the interval [ 1 , 3 ] .

  • A √ 3 2 βˆ’ 1 2
  • B 4 √ 5 βˆ’ 4
  • C √ 5 2 βˆ’ 1 2
  • D √ 5 βˆ’ 1

Q6:

Find ο„Έ 2 π‘₯ 9 + 2 π‘₯ π‘₯ 3 0 2 d to the nearest hundredth.

Q7:

Evaluate ο„Έ ( 2 βˆ’ π‘Ÿ ) π‘Ÿ 2 1 3 d .

  • A 1 1 3
  • B81
  • C βˆ’ 8 1
  • D 1 4
  • E βˆ’ 1 4

Q8:

Determine ο„Έ 8 π‘₯ βˆ’ 8 √ π‘₯ βˆ’ 1 π‘₯ 4 1 d to the nearest hundredth.

Q9:

Evaluate ο„Έ π‘₯ √ 5 π‘₯ + 3 π‘₯ 5 3 2 d to the nearest thousandth.

Q10:

Find to the nearest thousandth.

Q11:

Determine ο„Έ 4 π‘₯ ( π‘₯ βˆ’ 6 ) π‘₯ 7 5 6 d rounded to one decimal place.

Q12:

Find ο„Έ βˆ’ 9 ( 𝑧 ) ( 𝑧 ) 𝑧 πœ‹ 4 0 2 t a n s e c d .

  • A 9 2
  • B 1 2
  • C βˆ’ 9
  • D βˆ’ 9 2

Q13:

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

  • A l n l n 9 + ο€» 1 + √ 1 0 
  • B l n l n 1 0 βˆ’ ο€» 1 + √ 1 0 
  • C l n l n 3 βˆ’ ο€» 1 + √ 1 0 
  • D l n l n 9 βˆ’ ο€» 1 + √ 1 0 
  • E l n l n 3 + ο€» 1 βˆ’ √ 1 0 

Q14:

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

  • A l n ο€» 2 βˆ’ √ 2 
  • B l n ο€» 2 + √ 2 
  • C l n ( 2 )
  • D l n ο€» 1 + √ 2 
  • E l n ο€» √ 2 