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Lesson Worksheet: Antiderivatives Mathematics • 12th Grade

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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 𝐹(𝑥) of the function 𝑓(𝑥)=3𝑥−2𝑥−1.

  • A𝐹(𝑥)=3𝑥−2𝑥−𝑥+C
  • B𝐹(𝑥)=𝑥+𝑥−𝑥+C
  • C𝐹(𝑥)=𝑥+𝑥+𝑥+C
  • D𝐹(𝑥)=𝑥−𝑥−𝑥+C
  • E𝐹(𝑥)=9𝑥−4𝑥−𝑥+C

Q2:

Find the most general antiderivative 𝐹(𝑥) of the function 𝑓(𝑥)=2𝑥−3𝑥−𝑥.

  • A𝐹(𝑥)=𝑥+𝑥+𝑥+C
  • B𝐹(𝑥)=2𝑥−3𝑥−𝑥+C
  • C𝐹(𝑥)=16𝑥−18𝑥−3𝑥+C
  • D𝐹(𝑥)=𝑥4−𝑥2−𝑥3+C
  • E𝐹(𝑥)=𝑥8+𝑥6+𝑥3+C

Q3:

Determine the most general antiderivative 𝐹(𝑥) of the function 𝑓(𝑥)=−2√𝑥+3𝑥√𝑥.

  • A𝐹(𝑥)=−10𝑥3+6𝑥5+C
  • B𝐹(𝑥)=−6𝑥5+6𝑥5+C
  • C𝐹(𝑥)=−2𝑥+3𝑥+C
  • D𝐹(𝑥)=−3𝑥+9𝑥2+C
  • E𝐹(𝑥)=−2𝑥+3𝑥+C

Q4:

Find the most general antiderivative of the function 𝑓(𝑥)=4𝑥+3−23√𝑥sin.

  • A𝐹(𝑥)=−4√𝑥3+3𝑥−4𝑥+cosC
  • B𝐹(𝑥)=−2√𝑥3+3𝑥−4𝑥+cosC
  • C𝐹(𝑥)=4√𝑥+3𝑥−4𝑥+cosC
  • D𝐹(𝑥)=−2√𝑥3+3𝑥+4𝑥+cosC
  • E𝐹(𝑥)=−4√𝑥3+3𝑥+4𝑥+cosC

Q5:

If 𝐹(𝑥) and 𝐹(𝑥) are both antiderivatives of the same function 𝑓(𝑥), what is the relation between 𝐹 and 𝐹?

  • A𝐹 must be the double of 𝐹.
  • B𝐹−𝐹 is a constant.
  • C𝐹 must be equal to 𝐹.
  • D𝐹+𝐹 must be a constant.
  • EOnly one of them is undefined.

Q6:

By considering the product rule, find a function 𝑓 so that 𝑓′(𝑥)=𝑒√𝑥+2𝑒√𝑥.

  • A𝑓(𝑥)=2√𝑥𝑒+C
  • B𝑓(𝑥)=2𝑒√𝑥+C
  • C𝑓(𝑥)=√𝑥𝑒+C
  • D𝑓(𝑥)=𝑒√𝑥+C
  • E𝑓(𝑥)=√2𝑥𝑒+C

Q7:

If 𝑓′′(𝑥)=3𝑥+3𝑥+5𝑥+2, determine 𝑓(𝑥).

  • A𝑓(𝑥)=𝑥2+3𝑥4+5𝑥2+2𝑥+C
  • B𝑓(𝑥)=3𝑥+3𝑥+5𝑥+2𝑥+C
  • C𝑓(𝑥)=3𝑥+3𝑥+5𝑥+2𝑥+𝑥+CD
  • D𝑓(𝑥)=3𝑥4+3𝑥2+C
  • E𝑓(𝑥)=𝑥14+3𝑥20+5𝑥6+𝑥+𝑥+CD

Q8:

Determine 𝑓(𝑡) if 𝑓′′′(𝑡)=−4√𝑡+5𝑡cos.

  • A𝑓(𝑡)=−32𝑡105−5𝑡+𝑡+sinCD
  • B𝑓(𝑡)=−32𝑡105−5𝑡+𝑡+𝑡+sinCDE
  • C𝑓(𝑡)=−32𝑡105−5𝑡+𝑡+sinCE
  • D𝑓(𝑡)=−8𝑡3+5𝑡+sinC
  • E𝑓(𝑡)=−16𝑡15−5𝑡+𝑡+cosCD

Q9:

Determine the antiderivative 𝐹 of the function 𝑓(𝑥)=5𝑥+4𝑥 where 𝐹(1)=−2.

  • A𝐹(𝑥)=5𝑥+4𝑥−11
  • B𝐹(𝑥)=𝑥+𝑥−3
  • C𝐹(𝑥)=𝑥+𝑥−4
  • D𝐹(𝑥)=5𝑥+4𝑥+94
  • E𝐹(𝑥)=𝑥+𝑥+17

Q10:

Determine the function 𝑓 if 𝑓′(𝑥)=−3𝑥+1√𝑥 and 𝑓(1)=4.

  • A𝑓(𝑥)=−9𝑥2+2√𝑥+4
  • B𝑓(𝑥)=−2𝑥+2√𝑥+4
  • C𝑓(𝑥)=−2𝑥+2√𝑥+13
  • D𝑓(𝑥)=−9𝑥2+2√𝑥+13
  • E𝑓(𝑥))=−2𝑥+2√𝑥−4

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

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