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Lesson Worksheet: Antiderivatives Mathematics • 12th Grade

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:

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

Q2:

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

Q3:

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

Q4:

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

Q5:

If ๐น(๐‘ฅ)๏Šง and ๐น(๐‘ฅ)๏Šจ are both antiderivatives of the same function ๐‘“(๐‘ฅ), what is the relation between ๐น๏Šง and ๐น๏Šจ?

  • A๐น๏Šง must be the double of ๐น๏Šจ.
  • B๐นโˆ’๐น๏Šง๏Šจ is a constant.
  • C๐น๏Šง must be equal to ๐น๏Šจ.
  • D๐น+๐น๏Šง๏Šจ must be a constant.
  • EOnly one of them is undefined.

Q6:

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

Q7:

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

Q8:

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

Q9:

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

Q10:

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

This lesson includes 25 additional questions and 252 additional question variations for subscribers.

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