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Lesson Worksheet: Indefinite Integrals: Exponential and Reciprocal Functions Mathematics • Higher Education

In this worksheet, we will practice finding the indefinite integral of exponential and reciprocal functions (1/x).

Q1:

Determine ๏„ธ4๐œ‹๐‘’๐‘ฅ๏Šฉ๏—d.

  • A4๐œ‹3๐‘’+๏Šฉ๏—๏Šฑ๏ŠงC
  • B4๐œ‹๐‘’+๏Šฉ๏—C
  • C4๐œ‹3๐‘’+๏Šฉ๏—C
  • D4๐œ‹3๐‘’+๏Šฉ๏—๏Šฐ๏ŠงC

Q2:

Determine ๏„ธ๏€น8๐‘ฅ+7๐‘’๏…๐‘ฅ๏Šฉ๏Œพ๏Šฑ๏Šฎ๏—d.

  • A8๐‘ฅ3๐‘’+1+7๐‘’+๏Šฉ๏Œพ๏Šฐ๏Šง๏Šฑ๏Šฎ๏—C
  • B8๐‘ฅ3๐‘’+1โˆ’78๐‘’+๏Šฉ๏Œพ๏Šฐ๏Šง๏Šฑ๏Šฎ๏—C
  • Cโˆ’78๐‘’+83๐‘’๐‘ฅ+๏Šฑ๏Šฎ๏—๏Šฐ๏Šง๏Šฉ๏Œพ๏Šฐ๏ŠงC
  • Dโˆ’78๐‘’+83๐‘’๐‘ฅ+๏Šฑ๏Šฎ๏—๏Šฉ๏Œพ๏Šฐ๏ŠงC

Q3:

Determine๏„ธ2๐‘ฅ๏Šฏ๏—d.

  • A292+๏Šฏ๏—lnC
  • B22+๏Šฏ๏—lnC
  • C22+๏Šฏ๏—lnC
  • Dโˆ’292+๏Šฏ๏—lnC

Q4:

Determine ๏„ธโˆ’27๐‘ฅ๐‘ฅd.

  • Aโˆ’27|๐‘ฅ|+lnC
  • B27|๐‘ฅ|+lnC
  • C27|๐‘ฅ|+lnC
  • Dโˆ’27|๐‘ฅ|+lnC

Q5:

Determine ๏„ธ7๐‘ฅโˆ’36๐‘ฅ๐‘ฅ๏Šจd.

  • A7๐‘ฅ12โˆ’12|๐‘ฅ|+๏ŠจlnC
  • B7๐‘ฅ3โˆ’12|๐‘ฅ|+๏ŠจlnC
  • C7๐‘ฅ12โˆ’12|๐‘ฅ|+๏ŠจlnC
  • D7๐‘ฅ6โˆ’12|๐‘ฅ|+๏ŠจlnC

Q6:

Determine ๏„ธ8๐‘’โˆ’๐‘’+97๐‘’๐‘ฅ๏Šฉ๏—๏Šจ๏—๏—d.

  • A87๐‘’โˆ’๐‘’7+97๐‘’+๏Šจ๏—๏—๏Šฑ๏—C
  • B47๐‘’โˆ’๐‘’7+97๐‘’+๏Šจ๏—๏—๏Šฑ๏—C
  • C47๐‘’โˆ’๐‘’7โˆ’97๐‘’+๏Šจ๏—๏—๏Šฑ๏—C
  • D167๐‘’โˆ’๐‘’7โˆ’97๐‘’+๏Šจ๏—๏—๏Šฑ๏—C

Q7:

Determine ๏„ธ(โˆ’2๐‘ฅโˆ’5)๐‘ฅ๐‘ฅ๏Šจd.

  • A4๐‘ฅ+20๐‘ฅ+25|๐‘ฅ|+๏ŠจlnC
  • B4๐‘ฅ+20๐‘ฅ+25|๐‘ฅ|+๏ŠจlnC
  • C2๐‘ฅ+20๐‘ฅ+25|๐‘ฅ|+๏ŠจlnC
  • D2๐‘ฅ+20๐‘ฅ+25|๐‘ฅ|+๏ŠจlnC

Q8:

Determine ๏„ธ๏€ฟ3โˆš๐‘ฅ+79โˆš๐‘ฅ๏‹๐‘ฅ๏Šจd.

  • A9๐‘ฅ2+7๐‘ฅ3+4981|๐‘ฅ|+๏ŠจlnC
  • B9๐‘ฅ2+14๐‘ฅ3+4981|๐‘ฅ|+๏ŠจlnC
  • C9๐‘ฅ+14๐‘ฅ3+4981|๐‘ฅ|+๏ŠจlnC
  • D3๐‘ฅ2+14๐‘ฅ3+79|๐‘ฅ|+๏ŠจlnC

Q9:

Find, if possible, an antiderivative ๐น of ๐‘“(๐‘ฅ)=12๐‘ฅโˆ’1 that satisfies the conditions ๐น(0)=1 and ๐น(1)=โˆ’1.

  • ANo such antiderivative exists.
  • B๐น(๐‘ฅ)=โŽงโŽจโŽฉ12(1โˆ’2๐‘ฅ)+1๐‘ฅ<12,12(2๐‘ฅโˆ’1)โˆ’1๐‘ฅ>12lnforlnfor

Q10:

Given that ๐‘“โ€ฒโ€ฒ(๐‘ฅ)=โˆ’5๐‘’+2๐‘ฅ๏Šช๏—๏Šซ, find ๐‘“(๐‘ฅ).

  • A๐‘“(๐‘ฅ)=โˆ’54๐‘’+๐‘ฅ3+๏Šช๏—๏ŠฌC
  • B๐‘“(๐‘ฅ)=โˆ’5๐‘’+2๐‘ฅ+๐‘ฅ+๐ท๏Šช๏—๏ŠญC
  • C๐‘“(๐‘ฅ)=โˆ’516๐‘’+๐‘ฅ21+๐‘ฅ+๐ท๏Šช๏—๏ŠญC
  • D๐‘“(๐‘ฅ)=โˆ’5๐‘’+๐‘ฅ21+๐‘ฅ+๐ท๏Šฌ๏—๏ŠญC
  • E๐‘“(๐‘ฅ)=โˆ’5๐‘’+2๐‘ฅ+๏Šช๏—๏ŠฌC

This lesson includes 47 additional questions and 261 additional question variations for subscribers.

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