Worksheet: Antiderivatives

In this worksheet, we will practice finding the antiderivative of a function. The antiderivative of a function f(x) is the function F(x) where F′(x) = f(x).

Q1:

Determine the most general antiderivative ๐น(๐‘ฅ) of the function ๐‘“(๐‘ฅ)=โˆ’2๐‘’๏Šจ.

  • A๐น(๐‘ฅ)=โˆ’2๐‘’๐‘ฅ+๏ŠจC
  • B๐น(๐‘ฅ)=โˆ’2๐‘’๐‘ฅ๏Šจ
  • C๐น(๐‘ฅ)=โˆ’6๐‘’+๏ŠฉC
  • D๐น(๐‘ฅ)=โˆ’2๐‘’3๏Šฉ
  • E๐น(๐‘ฅ)=โˆ’2๐‘’3+๏ŠฉC

Q2:

Find the antiderivative of the function ๐‘“(๐‘ฅ)=2๐‘ฅ+3๐‘ฅ+3๏Šจ.

  • A2๐‘ฅ3+3๐‘ฅ+3๐‘ฅ+๏Šฉ๏ŠจC
  • B2๐‘ฅ+3๐‘ฅ+3๐‘ฅ+๏Šฉ๏ŠจC
  • C๐‘ฅ+3๐‘ฅ+3๐‘ฅ+๏Šฉ๏ŠจC
  • D2๐‘ฅ3+3๐‘ฅ2+3๐‘ฅ+๏Šฉ๏ŠจC
  • E2๐‘ฅ+3๐‘ฅ2+3๐‘ฅ+๏Šฉ๏ŠจC

Q3:

If ๐‘“โ€ฒโ€ฒ(๐‘ฅ)=3๐‘ฅ+3๐‘ฅ+5๐‘ฅ+2๏Šซ๏Šฉ, determine ๐‘“(๐‘ฅ).

  • A๐‘“(๐‘ฅ)=๐‘ฅ2+3๐‘ฅ4+5๐‘ฅ2+2๐‘ฅ+๏Šฌ๏Šช๏ŠจC
  • B๐‘“(๐‘ฅ)=3๐‘ฅ+3๐‘ฅ+5๐‘ฅ+2๐‘ฅ+๏Šญ๏Šซ๏Šฉ๏ŠจC
  • C๐‘“(๐‘ฅ)=3๐‘ฅ+3๐‘ฅ+5๐‘ฅ+2๐‘ฅ+๐‘ฅ+๏Šญ๏Šซ๏Šฉ๏ŠจCD
  • D๐‘“(๐‘ฅ)=3๐‘ฅ4+3๐‘ฅ2+๏Šช๏ŠจC
  • E๐‘“(๐‘ฅ)=๐‘ฅ14+3๐‘ฅ20+5๐‘ฅ6+๐‘ฅ+๐‘ฅ+๏Šญ๏Šซ๏Šฉ๏ŠจCD

Q4:

Find the most general antiderivative ๐น(๐‘ฅ) of the function ๐‘“(๐‘ฅ)=2๐‘ฅโˆ’3๐‘ฅโˆ’๐‘ฅ๏Šญ๏Šซ๏Šจ.

  • A๐น(๐‘ฅ)=๐‘ฅ+๐‘ฅ+๐‘ฅ+๏Šฎ๏Šฌ๏ŠฉC
  • B๐น(๐‘ฅ)=2๐‘ฅโˆ’3๐‘ฅโˆ’๐‘ฅ+๏Šฎ๏Šฌ๏ŠฉC
  • C๐น(๐‘ฅ)=16๐‘ฅโˆ’18๐‘ฅโˆ’3๐‘ฅ+๏Šฎ๏Šฌ๏ŠฉC
  • D๐น(๐‘ฅ)=๐‘ฅ4โˆ’๐‘ฅ2โˆ’๐‘ฅ3+๏Šฎ๏Šฌ๏ŠฉC
  • E๐น(๐‘ฅ)=๐‘ฅ8+๐‘ฅ6+๐‘ฅ3+๏Šฎ๏Šฌ๏ŠฉC

Q5:

Determine the most general antiderivative ๐น(๐‘ฅ) of the function ๐‘“(๐‘ฅ)=4๐‘ฅ(โˆ’๐‘ฅ+5).

  • A๐น(๐‘ฅ)=๐‘ฅโˆ’5๐‘ฅ+๏Šฉ๏ŠจC
  • B๐น(๐‘ฅ)=โˆ’4๐‘ฅโˆ’5๐‘ฅ+๏ŠจC
  • C๐น(๐‘ฅ)=โˆ’4๐‘ฅ3+10๐‘ฅ+๏ŠจC
  • D๐น(๐‘ฅ)=โˆ’4๐‘ฅ3+10๐‘ฅ+๏Šฉ๏ŠจC
  • E๐น(๐‘ฅ)=โˆ’4๐‘ฅ+20๐‘ฅ+๏Šฉ๏ŠจC

Q6:

Determine the antiderivative ๐น of the function ๐‘“(๐‘ฅ)=5๐‘ฅ+4๐‘ฅ๏Šช๏Šฉ where ๐น(1)=โˆ’2.

  • A๐น(๐‘ฅ)=5๐‘ฅ+4๐‘ฅโˆ’11๏Šซ๏Šช
  • B๐น(๐‘ฅ)=๐‘ฅ+๐‘ฅโˆ’3๏Šซ๏Šช
  • C๐น(๐‘ฅ)=๐‘ฅ+๐‘ฅโˆ’4๏Šซ๏Šช
  • D๐น(๐‘ฅ)=5๐‘ฅ+4๐‘ฅ+94๏Šซ๏Šช
  • E๐น(๐‘ฅ)=๐‘ฅ+๐‘ฅ+17๏Šซ๏Šช

Q7:

Determine the most general antiderivative ๐น(๐‘ฅ) of the function ๐‘“(๐‘ฅ)=6๐‘ฅ+7๐‘ฅ๏Ž ๏Žค๏Žก๏Žค๏Šฑ.

  • A๐น(๐‘ฅ)=๐‘ฅ+7๐‘ฅ3+๏Žฅ๏Žค๏Žข๏ŽคC
  • B๐น(๐‘ฅ)=๐‘ฅ+7๐‘ฅ3+๏Žค๏Žฅ๏Žค๏ŽขC
  • C๐น(๐‘ฅ)=36๐‘ฅ5+21๐‘ฅ5+๏Žค๏Žฅ๏Žค๏ŽขC
  • D๐น(๐‘ฅ)=5๐‘ฅ+35๐‘ฅ3+๏Žฅ๏Žค๏Žข๏ŽคC
  • E๐น(๐‘ฅ)=5๐‘ฅ6+5๐‘ฅ3+๏Žฅ๏Žค๏Žข๏ŽคC

Q8:

Find the most general antiderivative ๐น(๐‘ฅ) of the function ๐‘“(๐‘ฅ)=(๐‘ฅโˆ’3)๏Šจ.

  • A๐น(๐‘ฅ)=๐‘ฅ3โˆ’3๐‘ฅ+9๐‘ฅ+๏Šฉ๏ŠจC
  • B๐น(๐‘ฅ)=๐‘ฅ3โˆ’3๐‘ฅโˆ’3๐‘ฅ+๏Šฉ๏ŠจC
  • C๐น(๐‘ฅ)=๐‘ฅ+๐‘ฅ+9๐‘ฅ+๏Šฉ๏ŠจC
  • D๐น(๐‘ฅ)=๐‘ฅโˆ’3๐‘ฅ+9๐‘ฅ+๏Šฉ๏ŠจC
  • E๐น(๐‘ฅ)=3๐‘ฅโˆ’3๐‘ฅโˆ’3๐‘ฅ+๏Šฉ๏ŠจC

Q9:

Find the most general antiderivative ๐บ(๐‘ก) of the function ๐‘”(๐‘ก)=โˆ’3๐‘ก+5๐‘ก+44โˆš๐‘ก๏Šจ.

  • A๐บ(๐‘ก)=โˆ’3๐‘ก10โˆ’5๐‘ก6โˆ’2โˆš๐‘ก+๏Žค๏Žก๏Žข๏ŽกC
  • B๐บ(๐‘ก)=โˆ’3๐‘ก4โˆ’5๐‘ก4โˆ’โˆš๐‘ก+๏Žก๏Žค๏Žก๏ŽขC
  • C๐บ(๐‘ก)=โˆ’3๐‘ก4โˆ’5๐‘ก4โˆ’โˆš๐‘ก+๏Žค๏Žก๏Žข๏ŽกC
  • D๐บ(๐‘ก)=โˆ’๐‘ก10โˆ’๐‘ก6โˆ’โˆš๐‘ก2+๏Žค๏Žก๏Žข๏ŽกC
  • E๐บ(๐‘ก)=โˆ’15๐‘ก8โˆ’15๐‘ก8โˆ’2โˆš๐‘ก+๏Žก๏Žค๏Žก๏ŽขC

Q10:

Find the most general antiderivative of the function ๐‘“(๐‘ฅ)=4๐‘ฅ+3โˆ’23โˆš๐‘ฅsin.

  • A๐น(๐‘ฅ)=โˆ’4โˆš๐‘ฅ3+3๐‘ฅโˆ’4๐‘ฅ+cosC
  • B๐น(๐‘ฅ)=โˆ’2โˆš๐‘ฅ3+3๐‘ฅโˆ’4๐‘ฅ+cosC
  • C๐น(๐‘ฅ)=4โˆš๐‘ฅ+3๐‘ฅโˆ’4๐‘ฅ+cosC
  • D๐น(๐‘ฅ)=โˆ’2โˆš๐‘ฅ3+3๐‘ฅ+4๐‘ฅ+cosC
  • E๐น(๐‘ฅ)=โˆ’4โˆš๐‘ฅ3+3๐‘ฅ+4๐‘ฅ+cosC

Q11:

Determine the most general antiderivative ๐น(๐‘ฅ) of the function ๐‘“(๐‘ฅ)=โˆ’2โˆš๐‘ฅ+3๐‘ฅโˆš๐‘ฅ๏Žข๏Šจ.

  • A๐น(๐‘ฅ)=โˆ’10๐‘ฅ3+6๐‘ฅ5+๏Žค๏Žข๏Žค๏ŽกC
  • B๐น(๐‘ฅ)=โˆ’6๐‘ฅ5+6๐‘ฅ5+๏Žค๏Žข๏Žค๏ŽกC
  • C๐น(๐‘ฅ)=โˆ’2๐‘ฅ+3๐‘ฅ+๏Žข๏Žค๏Žก๏ŽคC
  • D๐น(๐‘ฅ)=โˆ’3๐‘ฅ+9๐‘ฅ2+๏Žข๏Žค๏Žก๏ŽคC
  • E๐น(๐‘ฅ)=โˆ’2๐‘ฅ+3๐‘ฅ+๏Žค๏Žข๏Žค๏ŽกC

Q12:

Determine the most general antiderivative ๐น(๐‘ฅ) of the function ๐‘“(๐‘ฅ)=โˆš๐‘ฅ+5โˆš๐‘ฅ๏Žข.

  • A๐น(๐‘ฅ)=4๐‘ฅ3+15๐‘ฅ2+๏Žฃ๏Žข๏Žข๏ŽกC
  • B๐น(๐‘ฅ)=3๐‘ฅ4+10๐‘ฅ3+๏Žฃ๏Žข๏Žข๏ŽกC
  • C๐น(๐‘ฅ)=๐‘ฅ+5๐‘ฅ+๏Žข๏Žฃ๏Žก๏ŽขC
  • D๐น(๐‘ฅ)=๐‘ฅ+5๐‘ฅ+๏Žฃ๏Žข๏Žข๏ŽกC
  • E๐น(๐‘ฅ)=4๐‘ฅ3+15๐‘ฅ2+๏Žข๏Žฃ๏Žก๏ŽขC

Q13:

Determine the most general antiderivative ๐น(๐‘ฅ) of the function ๐‘“ if ๐‘“(๐‘ฅ)=โˆ’2๐‘ฅ+๐‘ฅโˆ’2๐‘ฅ2๐‘ฅ๏Šช๏Šฉ๏Šฉ and ๐‘ฅ>0.

  • A๐น(๐‘ฅ)=โˆ’๐‘ฅ2+๐‘ฅ2+1๐‘ฅ+๏ŠจC, ๐‘ฅ>0
  • B๐น(๐‘ฅ)=โˆ’๐‘ฅ2+๐‘ฅ2โˆ’1๐‘ฅ+๏ŠจC, ๐‘ฅ>0
  • C๐น(๐‘ฅ)=โˆ’๐‘ฅ+๐‘ฅ2+1๐‘ฅ+๏ŠจC, ๐‘ฅ>0
  • D๐น(๐‘ฅ)=โˆ’๐‘ฅ2+๐‘ฅ2+2๐‘ฅ+๏ŠจC, ๐‘ฅ>0
  • E๐น(๐‘ฅ)=โˆ’2๐‘ฅ+๐‘ฅ2+1๐‘ฅ+๏ŠจC, ๐‘ฅ>0

Q14:

By considering the product rule, find a function ๐‘“ so that ๐‘“โ€ฒ(๐‘ฅ)=๐‘’โˆš๐‘ฅ+2๐‘’โˆš๐‘ฅ๏—๏—.

  • A๐‘“(๐‘ฅ)=2โˆš๐‘ฅ๐‘’+๐ถ๏—
  • B๐‘“(๐‘ฅ)=2๐‘’โˆš๐‘ฅ+๐ถ๏—
  • C๐‘“(๐‘ฅ)=โˆš๐‘ฅ๐‘’+๐ถ๏—
  • D๐‘“(๐‘ฅ)=๐‘’โˆš๐‘ฅ+๐ถ๏—
  • E๐‘“(๐‘ฅ)=โˆš2๐‘ฅ๐‘’+๐ถ๏—

Q15:

Find the most general antiderivative ๐น(๐‘ฅ) of the function ๐‘“(๐‘ฅ)=4๐‘ฅโˆ’2.

  • A๐น(๐‘ฅ)=๐‘ฅ2โˆ’2๐‘ฅ+๏ŠจC
  • B๐น(๐‘ฅ)=4๐‘ฅโˆ’2๐‘ฅ+๏ŠจC
  • C๐น(๐‘ฅ)=๐‘ฅโˆ’2๐‘ฅ+๏ŠจC
  • D๐น(๐‘ฅ)=8๐‘ฅโˆ’2๐‘ฅ+๏ŠจC
  • E๐น(๐‘ฅ)=2๐‘ฅโˆ’2๐‘ฅ+๏ŠจC

Q16:

Determine the family of functions ๐‘“ for which ๐‘“โ€ฒโ€ฒโ€ฒ(๐‘ก)=3๐‘กโˆ’2sin.

  • A๐‘“(๐‘ก)=โˆ’๐‘ก3+3๐‘ก+๐‘ก+๐‘ก+๏Šฉ๏ŠจcosCDE
  • B๐‘“(๐‘ก)=โˆ’2๐‘ก3+3๐‘ก+๐‘ก+๐‘ก+๏Šฉ๏ŠจcosCDE
  • C๐‘“(๐‘ก)=โˆ’๐‘ก3โˆ’3๐‘ก+๐‘ก+๐‘ก๏Šฉ๏ŠจcosCD
  • D๐‘“(๐‘ก)=โˆ’๐‘ก3+3๐‘ก+๐‘ก+๐‘ก๏Šฉ๏ŠจcosCD
  • E๐‘“(๐‘ก)=โˆ’๐‘ก3โˆ’3๐‘ก+๐‘ก+๐‘ก+๏Šฉ๏ŠจcosCDE

Q17:

Determine the function ๐‘“ if ๐‘“โ€ฒ(๐‘ฅ)=โˆ’3๐‘ฅ+1โˆš๐‘ฅ and ๐‘“(1)=4.

  • A๐‘“(๐‘ฅ)=โˆ’9๐‘ฅ2+2โˆš๐‘ฅ+4๏Žข๏Žก
  • B๐‘“(๐‘ฅ)=โˆ’2๐‘ฅ+2โˆš๐‘ฅ+4๏Žข๏Žก
  • C๐‘“(๐‘ฅ)=โˆ’2๐‘ฅ+2โˆš๐‘ฅ+13๏Žข๏Žก
  • D๐‘“(๐‘ฅ)=โˆ’9๐‘ฅ2+2โˆš๐‘ฅ+13๏Žข๏Žก
  • E๐‘“(๐‘ฅ))=โˆ’2๐‘ฅ+2โˆš๐‘ฅโˆ’4๏Žข๏Žก

Q18:

Determine ๐‘“(๐‘ก) if ๐‘“โ€ฒโ€ฒโ€ฒ(๐‘ก)=โˆ’4โˆš๐‘ก+5๐‘กcos.

  • A๐‘“(๐‘ก)=โˆ’32๐‘ก105โˆ’5๐‘ก+๐‘ก+๏Žฆ๏ŽกsinCD
  • B๐‘“(๐‘ก)=โˆ’32๐‘ก105โˆ’5๐‘ก+๐‘ก+๐‘ก+๏Žฆ๏ŽกsinCDE๏Šจ
  • C๐‘“(๐‘ก)=โˆ’32๐‘ก105โˆ’5๐‘ก+๐‘ก+๏Žฆ๏ŽกsinCE๏Šจ
  • D๐‘“(๐‘ก)=โˆ’8๐‘ก3+5๐‘ก+๏Žข๏ŽกsinC
  • E๐‘“(๐‘ก)=โˆ’16๐‘ก15โˆ’5๐‘ก+๐‘ก+๏Žค๏ŽกcosCD

Q19:

Find the most general antiderivative ๐น(๐‘ฅ) of the function ๐‘“(๐‘ฅ)=3๐‘ฅโˆ’2๐‘ฅโˆ’1๏Šจ.

  • A๐น(๐‘ฅ)=3๐‘ฅโˆ’2๐‘ฅโˆ’๐‘ฅ+๏Šฉ๏ŠจC
  • B๐น(๐‘ฅ)=๐‘ฅ+๐‘ฅโˆ’๐‘ฅ+๏Šฉ๏ŠจC
  • C๐น(๐‘ฅ)=๐‘ฅ+๐‘ฅ+๐‘ฅ+๏Šฉ๏ŠจC
  • D๐น(๐‘ฅ)=๐‘ฅโˆ’๐‘ฅโˆ’๐‘ฅ+๏Šฉ๏ŠจC
  • E๐น(๐‘ฅ)=9๐‘ฅโˆ’4๐‘ฅโˆ’๐‘ฅ+๏Šฉ๏ŠจC

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