Worksheets · Calculus

Inverse trig integrals worksheet

Two shapes are worth knowing on sight: one over a sum of squares gives an inverse tangent, and one over the root of a difference of squares gives an inverse sine. Everything else is getting the constants out front.

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Pick more than one for a sheet that mixes them.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Inverse Trig Integrals

Date Period

Evaluate each integral.

  1. ∫−436−25x2 dx
  2. ∫59−9x2 dx
  3. ∫−25+9x2 dx
  4. ∫−4−9x2 dx
  5. ∫−9−9x2 dx
  6. ∫−264−16x2 dx
  7. ∫−549−9x2 dx
  8. ∫−581+4x2 dx
  9. ∫−16−16x2 dx
  10. ∫−4−16x2 dx
  11. ∫59+9x2 dx
  12. ∫649−16x2 dx
  13. ∫164−4x2 dx
  14. ∫49−25x2 dx
  15. ∫64−4x2 dx
  16. ∫−4+9x2 dx
  17. ∫436−4x2 dx
  18. ∫−64−9x2 dx
  19. ∫−49+16x2 dx
  20. ∫−436+16x2 dx

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

Sigma Prep · sigmaprep.io/worksheets

Name

Inverse Trig IntegralsAnswer keyVersion 1

Date Period

Evaluate each integral.

  1. ∫−436−25x2 dx−45sin⁡−1(5x6)+C
  2. ∫59−9x2 dx53sin⁡−1(3x3)+C
  3. ∫−25+9x2 dx−115tan⁡−1(3x5)+C
  4. ∫−4−9x2 dx−13sin⁡−1(3x2)+C
  5. ∫−9−9x2 dx−13sin⁡−1(3x3)+C
  6. ∫−264−16x2 dx−12sin⁡−1(4x8)+C
  7. ∫−549−9x2 dx−53sin⁡−1(3x7)+C
  8. ∫−581+4x2 dx−518tan⁡−1(2x9)+C
  9. ∫−16−16x2 dx−14sin⁡−1(4x4)+C
  10. ∫−4−16x2 dx−14sin⁡−1(4x2)+C
  11. ∫59+9x2 dx59tan⁡−1(3x3)+C
  12. ∫649−16x2 dx32sin⁡−1(4x7)+C
  13. ∫164−4x2 dx12sin⁡−1(2x8)+C
  14. ∫49−25x2 dx45sin⁡−1(5x3)+C
  15. ∫64−4x2 dx3sin⁡−1(2x2)+C
  16. ∫−4+9x2 dx−16tan⁡−1(3x2)+C
  17. ∫436−4x2 dx2sin⁡−1(2x6)+C
  18. ∫−64−9x2 dx−13sin⁡−1(3x8)+C
  19. ∫−49+16x2 dx−13tan⁡−1(4x3)+C
  20. ∫−436+16x2 dx−16tan⁡−1(4x6)+C

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

How to do these

∫19+x2 dx

  1. This is one over a squared plus x squared, with a equal to 3.
  2. That integrates to one over a, times inverse tangent of x over a.
  3. So it is a third of inverse tangent of x over 3.

The answer is 13tan⁡−1(x3)+C.

Where students go wrong

Losing the one over a. The x over a inside is easy to remember; the matching one over a in front comes from the substitution and is what most of the lost marks are.

Also called arctan integral, arcsin integral, integrals giving inverse trig or a squared plus x squared.

Questions about these worksheets

Yes. Every inverse trig integrals 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 inverse trig integrals 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.

Losing the one over a. The x over a inside is easy to remember; the matching one over a in front comes from the substitution and is what most of the lost marks are.