Worksheets · Calculus

Antiderivatives with initial conditions worksheet

An antiderivative is only known up to a constant. One point on the graph fixes it. With a second derivative there are two constants, so it takes two conditions.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Antiderivatives With Initial Conditions

Date Period

Find the function that fits the derivative and the conditions given.

  1. f′(x)=9x2+4x+6 and f(1)=10. Find f(x)
  2. f′(x)=− ⁣sin⁡3x and f(0)=−1. Find f(x)
  3. f′′(x)=−12x2−8, f′(1)=−17 and f(1)=−14. Find f(x)
  4. A particle moves along a line with velocity v(t)=9t2−4t−6 and s(2)=7. Find its position s(t).
  5. f′(x)=−9x2−8x+2 and f(0)=5. Find f(x)
  6. f′(x)=−2x−6x for x>0 and f(1)=−3. Find f(x)
  7. f′′(x)=12x2−12x−8, f(0)=−5 and f(−1)=−12. Find f(x)
  8. A particle moves along a line with velocity v(t)=−9t2+10t+8 and s(3)=−10. Find its position s(t).
  9. f′(x)=3x2−10x−7 and f(2)=−26. Find f(x)
  10. f′(x)=−5sin⁡4x and f(0)=4. Find f(x)
  11. f′′(x)=−24x2−12x−2, f(0)=−3 and f(2)=−61. Find f(x)
  12. A particle moves along a line with velocity v(t)=3t2−8t+3 and s(2)=−3. Find its position s(t).
  13. f′(x)=−3x2−8x+1 and f(−1)=−4. Find f(x)
  14. f′(x)=2sec⁡2x and f(π/4)=−5. Find f(x)
  15. f′′(x)=24x2+18x−8, f′(2)=83 and f(2)=47. Find f(x)
  16. A particle moves along a line with velocity v(t)=−3t2+8t+6 and s(3)=43. Find its position s(t).
  17. f′(x)=−6x2+4x+3 and f(−1)=9. Find f(x)
  18. f′(x)=6sin⁡2x and f(0)=3. Find f(x)
  19. f′′(x)=24x2−18x+6, f′(1)=9 and f(1)=4. Find f(x)
  20. A particle moves along a line with velocity v(t)=−6t2−2t−6 and s(1)=0. Find its position s(t).

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Antiderivatives With Initial ConditionsAnswer keyVersion 1

Date Period

Find the function that fits the derivative and the conditions given.

  1. f′(x)=9x2+4x+6 and f(1)=10. Find f(x)f(x)=3x3+2x2+6x−1
  2. f′(x)=− ⁣sin⁡3x and f(0)=−1. Find f(x)f(x)=13cos⁡3x−43
  3. f′′(x)=−12x2−8, f′(1)=−17 and f(1)=−14. Find f(x)f(x)=−x4−4x2−5x−4
  4. A particle moves along a line with velocity v(t)=9t2−4t−6 and s(2)=7. Find its position s(t).s(t)=3t3−2t2−6t+3
  5. f′(x)=−9x2−8x+2 and f(0)=5. Find f(x)f(x)=−3x3−4x2+2x+5
  6. f′(x)=−2x−6x for x>0 and f(1)=−3. Find f(x)f(x)=−2ln⁡x−3x2
  7. f′′(x)=12x2−12x−8, f(0)=−5 and f(−1)=−12. Find f(x)f(x)=x4−2x3−4x2+6x−5
  8. A particle moves along a line with velocity v(t)=−9t2+10t+8 and s(3)=−10. Find its position s(t).s(t)=−3t3+5t2+8t+2
  9. f′(x)=3x2−10x−7 and f(2)=−26. Find f(x)f(x)=x3−5x2−7x
  10. f′(x)=−5sin⁡4x and f(0)=4. Find f(x)f(x)=54cos⁡4x+114
  11. f′′(x)=−24x2−12x−2, f(0)=−3 and f(2)=−61. Find f(x)f(x)=−2x4−2x3−x2−3x−3
  12. A particle moves along a line with velocity v(t)=3t2−8t+3 and s(2)=−3. Find its position s(t).s(t)=t3−4t2+3t−1
  13. f′(x)=−3x2−8x+1 and f(−1)=−4. Find f(x)f(x)=−x3−4x2+x
  14. f′(x)=2sec⁡2x and f(π/4)=−5. Find f(x)f(x)=2tan⁡x−7
  15. f′′(x)=24x2+18x−8, f′(2)=83 and f(2)=47. Find f(x)f(x)=2x4+3x3−4x2−x+9
  16. A particle moves along a line with velocity v(t)=−3t2+8t+6 and s(3)=43. Find its position s(t).s(t)=−t3+4t2+6t+16
  17. f′(x)=−6x2+4x+3 and f(−1)=9. Find f(x)f(x)=−2x3+2x2+3x+8
  18. f′(x)=6sin⁡2x and f(0)=3. Find f(x)f(x)=−3cos⁡2x+6
  19. f′′(x)=24x2−18x+6, f′(1)=9 and f(1)=4. Find f(x)f(x)=2x4−3x3+3x2+4x−2
  20. A particle moves along a line with velocity v(t)=−6t2−2t−6 and s(1)=0. Find its position s(t).s(t)=−2t3−t2−6t+9

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

f′′(x)=6x−4, f′(0)=3 and f(0)=5. Find f(x)

  1. Integrate once: f'(x) = 3x^2 - 4x + C_1. Since f'(0) = 3, C_1 = 3.
  2. Integrate again: f(x) = x^3 - 2x^2 + 3x + C_2. Since f(0) = 5, C_2 = 5.

The answer is f(x)=x3−2x2+3x+5.

Where students go wrong

Using the condition on f' to find the constant in f. Each condition belongs to its own integration: f'(0) fixes the first constant, f(0) the second.

Also called initial value problems, particular antiderivative, finding f from f prime, antiderivative with a given point or particular solution of a differential equation.

What you can put on this worksheet

Find f from f' and a point
f′(x)=3x2−6x−7 and f(0)=−3. Find f(x) → f(x)=x3−3x2−7x−3
Trig, exponential and 1/x, with a condition
f′(x)=sin⁡x−6x and f(0)=1. Find f(x) → f(x)=− ⁣cos⁡x−3x2+2
f'' and two conditions
f′′(x)=12x+4, f′(0)=−3 and f(0)=1. Find f(x) → f(x)=2x3+2x2−3x+1
Position from velocity and a starting point
A particle moves along a line with velocity v(t)=8t−7 and s(0)=6. Find its position s(t). → s(t)=4t2−7t+6
Trig derivatives with a condition at a multiple of pi
Find f(x) if f′(x)=5cos⁡x and f(π)=−8 → f(x)=5sin⁡x−8
Reciprocal powers, shifted
Find the general solution of dydx=5x3 → y=−52x2+C

Questions about these worksheets

Yes. Every antiderivatives with initial conditions sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the antiderivatives with initial conditions questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 6: find f from f' and a point, trig, exponential and 1/x, with a condition, f'' and two conditions, position from velocity and a starting point, trig derivatives with a condition at a multiple of pi and reciprocal powers, shifted. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the condition on f' to find the constant in f. Each condition belongs to its own integration: f'(0) fixes the first constant, f(0) the second.