Worksheet: Integration by Substitution: Definite Integrals

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 ๏„ธ๐‘ฆ๐‘ฆโˆš1+(๐‘ฆ)๏Œพ๏Šง๏Šจdln.

  • Aln๏€ป2+โˆš2๏‡
  • Bln๏€ป2โˆ’โˆš2๏‡
  • Cln๏€ปโˆš2๏‡
  • Dln๏€ป1+โˆš2๏‡
  • Eln(2)

Q2:

Evaluate ๏„ธ(2โˆ’๐‘Ÿ)๐‘Ÿ๏Šจ๏Šง๏Šฉd.

  • A113
  • B14
  • C81
  • Dโˆ’81
  • Eโˆ’14

Q3:

Determine ๏„ธ8๐‘ฅโˆ’8โˆš๐‘ฅโˆ’1๐‘ฅ๏Šช๏Šงd to the nearest hundredth.

Q4:

Evaluate ๏„ธ๐‘ฅโˆš5๐‘ฅ+3๐‘ฅ๏Šซ๏Šฉ๏Šจd to the nearest thousandth.

Q5:

Find ๏„ธ๐‘ฅโˆš๐‘ฅ+5๐‘ฅ๏Šช๏Šฑ๏Šงd to the nearest thousandth.

Q6:

Determine ๏„ธ4๐‘ฅ(๐‘ฅโˆ’6)๐‘ฅ๏Šญ๏Šซ๏Šฌd rounded to one decimal place.

Q7:

Find ๏„ธโˆ’9(๐‘ง)(๐‘ง)๐‘ง๏‘ฝ๏Žฃ๏Šฆ๏Šจtansecd.

  • A92
  • Bโˆ’92
  • Cโˆ’9
  • D12

Q8:

Determine ๏„ธ2โˆš๐‘ฅ+6๐‘ฅ๏Šฑ๏Šจ๏Šฑ๏Šซd.

  • A12
  • B43
  • Cโˆ’4
  • D4

Q9:

Determine ๏„ธ5โˆšโˆ’๐‘ฅ+7๐‘ฅ๏Šฆ๏Šจd.

  • Aโˆ’50โˆš7+50โˆš5
  • Bโˆ’10โˆš7+10โˆš5
  • Cโˆ’20โˆš7+20โˆš5
  • Dโˆ’10โˆš5+10โˆš7

Q10:

Evaluate ๏„ธ๐‘ฅ๏€น๐‘ฅโˆ’3๏…๐‘ฅ๏Šจ๏Šง๏Šจ๏Šฉ๏Šฉd.

Q11:

Evaluate ๏„ธ8๐‘ก10๐‘ก+5๐‘ก๏Šจ๏Šง๏Šจd.

  • A25(27)ln
  • B25๏€ผ13๏ˆln
  • C85(3)ln
  • D25(3)ln
  • E85(27)ln

Q12:

Evaluate ๏„ธ๐‘’๐œƒ๐œƒ๏Ž„๏Šจ๏ผ๏‘ฝ๏Žขsincosd.

  • A12๏€ป๐‘’โˆ’1๏‡โˆš๏Šฉ
  • B2๏€ป๐‘’โˆ’1๏‡โˆš๏Šฉ
  • C๐‘’2โˆš๏Šฉ
  • D12๏€ป1โˆ’๐‘’๏‡โˆš๏Šฉ
  • E2๏€ป1โˆ’๐‘’๏‡โˆš๏Šฉ

Q13:

Evaluate ๏„ธ2๏€ป4โˆ’๐‘ฅ6๏‡๏€ป4โˆ’๐‘ฅ6๏‡๏Šจ๏Šช๏Šจ๏Šช๏Šฑ๏Ž„tansec, giving an exact answer.

  • A8โˆš3โˆ’12
  • B6โˆš3โˆ’12
  • C6โˆš3
  • D12โˆ’8โˆš3
  • E8โˆš3

Q14:

Evaluate ๏„ธ5๐‘ฅโˆ’7๏€บ๏€บ(๐‘ฅโˆ’7)๏†+1๏†๐‘ฅ๏Šญ๏Šฐ๏Œพ๏Šฎ๏Šจ๏Šฉlnd.

  • A50
  • B51.25
  • C2053
  • D10
  • E20,900๐‘’

Q15:

Evaluate ๏„ธ4๐‘ฅ๏€ป๏€ป๐œ‹6๐‘ฅ๏‡๏‡๐‘ฅ๏Šฉ๏Šฆ๏Šจ๏Šฉsind.

  • A8๐œ‹
  • B16๐œ‹
  • Cโˆ’8๐œ‹
  • Dโˆ’8๐œ‹
  • E8๐œ‹

Q16:

Determine ๏„ธ๐‘ฅ๐‘ฅ๐‘ฅ๏Œพ๏Šงlnd.

  • A14
  • B2
  • C1
  • D4
  • E12

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