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.

  • A ๐น ( ๐‘ฅ ) = โˆ’ 6 ๐‘’ + ๏Šฉ C
  • B ๐น ( ๐‘ฅ ) = โˆ’ 2 ๐‘’ ๐‘ฅ + ๏Šจ C
  • C ๐น ( ๐‘ฅ ) = โˆ’ 2 ๐‘’ ๐‘ฅ ๏Šจ
  • D ๐น ( ๐‘ฅ ) = โˆ’ 2 ๐‘’ 3 ๏Šฉ
  • E ๐น ( ๐‘ฅ ) = โˆ’ 2 ๐‘’ 3 + ๏Šฉ C

Q2:

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

  • A ๐‘ฅ + 3 ๐‘ฅ + 3 ๐‘ฅ + ๏Šฉ ๏Šจ C
  • B 2 ๐‘ฅ + 3 ๐‘ฅ + 3 ๐‘ฅ + ๏Šฉ ๏Šจ C
  • C 2 ๐‘ฅ + 3 ๐‘ฅ 2 + 3 ๐‘ฅ + ๏Šฉ ๏Šจ C
  • D 2 ๐‘ฅ 3 + 3 ๐‘ฅ + 3 ๐‘ฅ + ๏Šฉ ๏Šจ C
  • E 2 ๐‘ฅ 3 + 3 ๐‘ฅ 2 + 3 ๐‘ฅ + ๏Šฉ ๏Šจ C

Q3:

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

  • A ๐‘“ ( ๐‘ฅ ) = 3 ๐‘ฅ + 3 ๐‘ฅ + 5 ๐‘ฅ + 2 ๐‘ฅ + ๏Šญ ๏Šซ ๏Šฉ ๏Šจ C
  • B ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ 2 + 3 ๐‘ฅ 4 + 5 ๐‘ฅ 2 + 2 ๐‘ฅ + ๏Šฌ ๏Šช ๏Šจ C
  • C ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ 1 4 + 3 ๐‘ฅ 2 0 + 5 ๐‘ฅ 6 + ๐‘ฅ + ๐‘ฅ + ๏Šญ ๏Šซ ๏Šฉ ๏Šจ C D
  • D ๐‘“ ( ๐‘ฅ ) = 3 ๐‘ฅ + 3 ๐‘ฅ + 5 ๐‘ฅ + 2 ๐‘ฅ + ๐‘ฅ + ๏Šญ ๏Šซ ๏Šฉ ๏Šจ C D
  • E ๐‘“ ( ๐‘ฅ ) = 3 ๐‘ฅ 4 + 3 ๐‘ฅ 2 + ๏Šช ๏Šจ C

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.

  • A ๐น ( ๐‘ฅ ) = ๐‘ฅ + ๐‘ฅ + ๐‘ฅ + ๏Šฎ ๏Šฌ ๏Šฉ C
  • B ๐น ( ๐‘ฅ ) = 1 6 ๐‘ฅ โˆ’ 1 8 ๐‘ฅ โˆ’ 3 ๐‘ฅ + ๏Šฎ ๏Šฌ ๏Šฉ C
  • C ๐น ( ๐‘ฅ ) = ๐‘ฅ 8 + ๐‘ฅ 6 + ๐‘ฅ 3 + ๏Šฎ ๏Šฌ ๏Šฉ C
  • D ๐น ( ๐‘ฅ ) = 2 ๐‘ฅ โˆ’ 3 ๐‘ฅ โˆ’ ๐‘ฅ + ๏Šฎ ๏Šฌ ๏Šฉ C
  • E ๐น ( ๐‘ฅ ) = ๐‘ฅ 4 โˆ’ ๐‘ฅ 2 โˆ’ ๐‘ฅ 3 + ๏Šฎ ๏Šฌ ๏Šฉ C

Q5:

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

  • A ๐น ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 5 ๐‘ฅ + ๏Šฉ ๏Šจ C
  • B ๐น ( ๐‘ฅ ) = โˆ’ 4 ๐‘ฅ + 2 0 ๐‘ฅ + ๏Šฉ ๏Šจ C
  • C ๐น ( ๐‘ฅ ) = โˆ’ 4 ๐‘ฅ โˆ’ 5 ๐‘ฅ + ๏Šจ C
  • D ๐น ( ๐‘ฅ ) = โˆ’ 4 ๐‘ฅ 3 + 1 0 ๐‘ฅ + ๏Šจ C
  • E ๐น ( ๐‘ฅ ) = โˆ’ 4 ๐‘ฅ 3 + 1 0 ๐‘ฅ + ๏Šฉ ๏Šจ C

Q6:

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

  • A ๐น ( ๐‘ฅ ) = 5 ๐‘ฅ + 4 ๐‘ฅ โˆ’ 1 1 ๏Šซ ๏Šช
  • B ๐น ( ๐‘ฅ ) = ๐‘ฅ + ๐‘ฅ โˆ’ 4 ๏Šซ ๏Šช
  • C ๐น ( ๐‘ฅ ) = 5 ๐‘ฅ + 4 ๐‘ฅ + 9 4 ๏Šซ ๏Šช
  • D ๐น ( ๐‘ฅ ) = ๐‘ฅ + ๐‘ฅ + 1 7 ๏Šซ ๏Šช
  • E ๐น ( ๐‘ฅ ) = ๐‘ฅ + ๐‘ฅ โˆ’ 3 ๏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.

  • A ๐น ( ๐‘ฅ ) = ๐‘ฅ + 7 ๐‘ฅ 3 + ๏Žฅ ๏Žค ๏Žข ๏Žค C
  • B ๐น ( ๐‘ฅ ) = 5 ๐‘ฅ 6 + 5 ๐‘ฅ 3 + ๏Žฅ ๏Žค ๏Žข ๏Žค C
  • C ๐น ( ๐‘ฅ ) = ๐‘ฅ + 7 ๐‘ฅ 3 + ๏Žค ๏Žฅ ๏Žค ๏Žข C
  • D ๐น ( ๐‘ฅ ) = 3 6 ๐‘ฅ 5 + 2 1 ๐‘ฅ 5 + ๏Žค ๏Žฅ ๏Žค ๏Žข C
  • E ๐น ( ๐‘ฅ ) = 5 ๐‘ฅ + 3 5 ๐‘ฅ 3 + ๏Žฅ ๏Žค ๏Žข ๏Žค C

Q8:

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

  • A ๐น ( ๐‘ฅ ) = ๐‘ฅ + ๐‘ฅ + 9 ๐‘ฅ + ๏Šฉ ๏Šจ C
  • B ๐น ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 3 ๐‘ฅ + 9 ๐‘ฅ + ๏Šฉ ๏Šจ C
  • C ๐น ( ๐‘ฅ ) = ๐‘ฅ 3 โˆ’ 3 ๐‘ฅ โˆ’ 3 ๐‘ฅ + ๏Šฉ ๏Šจ C
  • D ๐น ( ๐‘ฅ ) = 3 ๐‘ฅ โˆ’ 3 ๐‘ฅ โˆ’ 3 ๐‘ฅ + ๏Šฉ ๏Šจ C
  • E ๐น ( ๐‘ฅ ) = ๐‘ฅ 3 โˆ’ 3 ๐‘ฅ + 9 ๐‘ฅ + ๏Šฉ ๏Šจ C

Q9:

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

  • A ๐บ ( ๐‘ก ) = โˆ’ ๐‘ก 1 0 โˆ’ ๐‘ก 6 โˆ’ โˆš ๐‘ก 2 + ๏Žค ๏Žก ๏Žข ๏Žก C
  • B ๐บ ( ๐‘ก ) = โˆ’ 1 5 ๐‘ก 8 โˆ’ 1 5 ๐‘ก 8 โˆ’ 2 โˆš ๐‘ก + ๏Žก ๏Žค ๏Žก ๏Žข C
  • C ๐บ ( ๐‘ก ) = โˆ’ 3 ๐‘ก 4 โˆ’ 5 ๐‘ก 4 โˆ’ โˆš ๐‘ก + ๏Žค ๏Žก ๏Žข ๏Žก C
  • D ๐บ ( ๐‘ก ) = โˆ’ 3 ๐‘ก 4 โˆ’ 5 ๐‘ก 4 โˆ’ โˆš ๐‘ก + ๏Žก ๏Žค ๏Žก ๏Žข C
  • E ๐บ ( ๐‘ก ) = โˆ’ 3 ๐‘ก 1 0 โˆ’ 5 ๐‘ก 6 โˆ’ 2 โˆš ๐‘ก + ๏Žค ๏Žก ๏Žข ๏Žก C

Q10:

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

  • A ๐น ( ๐‘ฅ ) = โˆ’ 2 โˆš ๐‘ฅ 3 + 3 ๐‘ฅ โˆ’ 4 ๐‘ฅ + c o s C
  • B ๐น ( ๐‘ฅ ) = โˆ’ 4 โˆš ๐‘ฅ 3 + 3 ๐‘ฅ + 4 ๐‘ฅ + c o s C
  • C ๐น ( ๐‘ฅ ) = โˆ’ 4 โˆš ๐‘ฅ 3 + 3 ๐‘ฅ โˆ’ 4 ๐‘ฅ + c o s C
  • D ๐น ( ๐‘ฅ ) = โˆ’ 2 โˆš ๐‘ฅ 3 + 3 ๐‘ฅ + 4 ๐‘ฅ + c o s C
  • E ๐น ( ๐‘ฅ ) = 4 โˆš ๐‘ฅ + 3 ๐‘ฅ โˆ’ 4 ๐‘ฅ + c o s C

Q11:

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

  • A ๐น ( ๐‘ฅ ) = โˆ’ 2 ๐‘ฅ + 3 ๐‘ฅ + ๏Žค ๏Žข ๏Žค ๏Žก C
  • B ๐น ( ๐‘ฅ ) = โˆ’ 2 ๐‘ฅ + 3 ๐‘ฅ + ๏Žข ๏Žค ๏Žก ๏Žค C
  • C ๐น ( ๐‘ฅ ) = โˆ’ 6 ๐‘ฅ 5 + 6 ๐‘ฅ 5 + ๏Žค ๏Žข ๏Žค ๏Žก C
  • D ๐น ( ๐‘ฅ ) = โˆ’ 1 0 ๐‘ฅ 3 + 6 ๐‘ฅ 5 + ๏Žค ๏Žข ๏Žค ๏Žก C
  • E ๐น ( ๐‘ฅ ) = โˆ’ 3 ๐‘ฅ + 9 ๐‘ฅ 2 + ๏Žข ๏Žค ๏Žก ๏Žค C

Q12:

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

  • A ๐น ( ๐‘ฅ ) = 3 ๐‘ฅ 4 + 1 0 ๐‘ฅ 3 + ๏Žฃ ๏Žข ๏Žข ๏Žก C
  • B ๐น ( ๐‘ฅ ) = ๐‘ฅ + 5 ๐‘ฅ + ๏Žฃ ๏Žข ๏Žข ๏Žก C
  • C ๐น ( ๐‘ฅ ) = 4 ๐‘ฅ 3 + 1 5 ๐‘ฅ 2 + ๏Žข ๏Žฃ ๏Žก ๏Žข C
  • D ๐น ( ๐‘ฅ ) = ๐‘ฅ + 5 ๐‘ฅ + ๏Žข ๏Žฃ ๏Žก ๏Žข C
  • E ๐น ( ๐‘ฅ ) = 4 ๐‘ฅ 3 + 1 5 ๐‘ฅ 2 + ๏Žฃ ๏Žข ๏Žข ๏Žก C

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.

  • A ๐น ( ๐‘ฅ ) = โˆ’ ๐‘ฅ 2 + ๐‘ฅ 2 + 1 ๐‘ฅ + ๏Šจ C , ๐‘ฅ > 0
  • B ๐น ( ๐‘ฅ ) = โˆ’ ๐‘ฅ + ๐‘ฅ 2 + 1 ๐‘ฅ + ๏Šจ C , ๐‘ฅ > 0
  • C ๐น ( ๐‘ฅ ) = โˆ’ ๐‘ฅ 2 + ๐‘ฅ 2 + 2 ๐‘ฅ + ๏Šจ C , ๐‘ฅ > 0
  • D ๐น ( ๐‘ฅ ) = โˆ’ 2 ๐‘ฅ + ๐‘ฅ 2 + 1 ๐‘ฅ + ๏Šจ C , ๐‘ฅ > 0
  • E ๐น ( ๐‘ฅ ) = โˆ’ ๐‘ฅ 2 + ๐‘ฅ 2 โˆ’ 1 ๐‘ฅ + ๏Šจ C , 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๏—๏—.

  • A ๐‘“ ( ๐‘ฅ ) = 2 ๐‘’ โˆš ๐‘ฅ ๏—
  • B ๐‘“ ( ๐‘ฅ ) = โˆš ๐‘ฅ ๐‘’ ๏—
  • C ๐‘“ ( ๐‘ฅ ) = ๐‘’ โˆš ๐‘ฅ ๏—
  • D ๐‘“ ( ๐‘ฅ ) = 2 โˆš ๐‘ฅ ๐‘’ ๏—
  • E ๐‘“ ( ๐‘ฅ ) = โˆš 2 ๐‘ฅ ๐‘’ ๏—

Q15:

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

  • A ๐น ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 2 ๐‘ฅ + ๏Šจ C
  • B ๐น ( ๐‘ฅ ) = 8 ๐‘ฅ โˆ’ 2 ๐‘ฅ + ๏Šจ C
  • C ๐น ( ๐‘ฅ ) = 2 ๐‘ฅ โˆ’ 2 ๐‘ฅ + ๏Šจ C
  • D ๐น ( ๐‘ฅ ) = ๐‘ฅ 2 โˆ’ 2 ๐‘ฅ + ๏Šจ C
  • E ๐น ( ๐‘ฅ ) = 4 ๐‘ฅ โˆ’ 2 ๐‘ฅ + ๏Šจ C

Q16:

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

  • A ๐‘“ ( ๐‘ก ) = โˆ’ ๐‘ก 3 โˆ’ 3 ๐‘ก + ๐‘ก + ๐‘ก + ๏Šฉ ๏Šจ c o s C D E
  • B ๐‘“ ( ๐‘ก ) = โˆ’ ๐‘ก 3 โˆ’ 3 ๐‘ก + ๐‘ก + ๐‘ก ๏Šฉ ๏Šจ c o s C D
  • C ๐‘“ ( ๐‘ก ) = โˆ’ ๐‘ก 3 + 3 ๐‘ก + ๐‘ก + ๐‘ก + ๏Šฉ ๏Šจ c o s C D E
  • D ๐‘“ ( ๐‘ก ) = โˆ’ 2 ๐‘ก 3 + 3 ๐‘ก + ๐‘ก + ๐‘ก + ๏Šฉ ๏Šจ c o s C D E
  • E ๐‘“ ( ๐‘ก ) = โˆ’ ๐‘ก 3 + 3 ๐‘ก + ๐‘ก + ๐‘ก ๏Šฉ ๏Šจ c o s C D

Q17:

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

  • A ๐‘“ ( ๐‘ฅ ) = โˆ’ 2 ๐‘ฅ + 2 โˆš ๐‘ฅ + 4 ๏Žข ๏Žก
  • B ๐‘“ ( ๐‘ฅ ) = โˆ’ 9 ๐‘ฅ 2 + 2 โˆš ๐‘ฅ + 1 3 ๏Žข ๏Žก
  • C ๐‘“ ( ๐‘ฅ ) ) = โˆ’ 2 ๐‘ฅ + 2 โˆš ๐‘ฅ โˆ’ 4 ๏Žข ๏Žก
  • D ๐‘“ ( ๐‘ฅ ) = โˆ’ 2 ๐‘ฅ + 2 โˆš ๐‘ฅ + 1 3 ๏Žข ๏Žก
  • E ๐‘“ ( ๐‘ฅ ) = โˆ’ 9 ๐‘ฅ 2 + 2 โˆš ๐‘ฅ + 4 ๏Žข ๏Žก

Q18:

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

  • A ๐‘“ ( ๐‘ก ) = โˆ’ 8 ๐‘ก 3 + 5 ๐‘ก + ๏Žข ๏Žก s i n C
  • B ๐‘“ ( ๐‘ก ) = โˆ’ 3 2 ๐‘ก 1 0 5 โˆ’ 5 ๐‘ก + ๐‘ก + ๏Žฆ ๏Žก s i n C E ๏Šจ
  • C ๐‘“ ( ๐‘ก ) = โˆ’ 3 2 ๐‘ก 1 0 5 โˆ’ 5 ๐‘ก + ๐‘ก + ๐‘ก + ๏Žฆ ๏Žก s i n C D E ๏Šจ
  • D ๐‘“ ( ๐‘ก ) = โˆ’ 3 2 ๐‘ก 1 0 5 โˆ’ 5 ๐‘ก + ๐‘ก + ๏Žฆ ๏Žก s i n C D
  • E ๐‘“ ( ๐‘ก ) = โˆ’ 1 6 ๐‘ก 1 5 โˆ’ 5 ๐‘ก + ๐‘ก + ๏Žค ๏Žก c o s C D

Q19:

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

  • A ๐น ( ๐‘ฅ ) = 3 ๐‘ฅ โˆ’ 2 ๐‘ฅ โˆ’ ๐‘ฅ + ๏Šฉ ๏Šจ C
  • B ๐น ( ๐‘ฅ ) = ๐‘ฅ + ๐‘ฅ โˆ’ ๐‘ฅ + ๏Šฉ ๏Šจ C
  • C ๐น ( ๐‘ฅ ) = ๐‘ฅ + ๐‘ฅ + ๐‘ฅ + ๏Šฉ ๏Šจ C
  • D ๐น ( ๐‘ฅ ) = ๐‘ฅ โˆ’ ๐‘ฅ โˆ’ ๐‘ฅ + ๏Šฉ ๏Šจ C
  • E ๐น ( ๐‘ฅ ) = 9 ๐‘ฅ โˆ’ 4 ๐‘ฅ โˆ’ ๐‘ฅ + ๏Šฉ ๏Šจ C

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