Skip to main content
Search IntMath

Table of Common Integrals

Leibniz's table of derivatives and integrals

Leibniz table of integrals
A simple table of derivatives and integrals from the Gottfried Leibniz archive. Leibniz developed integral calculus at around the same time as Isaac Newton. [Image source]

You can see how to use this table of common integrals in the previous section: Integration by Use of Tables.

1. `int1/(ax+b)dx` `=1/aln\ |ax+b|+K`

2. `int1/((ax+b)^2)dx` `=-1/(a(ax+b))+K`

3. `int1/((ax+b)^n)dx` `=-1/(a(n-1)(ax+b)^(n-1))+K`

4. `int1/(a^2+x^2)dx` `=1/atan^(-1)(x/a)+K`

Or, equivalently: `int1/(a^2+x^2)dx` `=1/a arctan (x/a)+K`

5. `int(f’(x))/(f(x))dx` `=ln\ |f(x)|+K`

6. `intsin^2udu` `=u/2-1/2sin u\ cos u+K`

7. `intsin^3udu` `=-cos u+1/3cos^3u+K`

8. `intsin^(n)u\ du` `=-1/nsin^(n-1)u\ cos u` `+(n-1)/nintsin^(n-2)u\ du`

9. `intcos^2u\ du` `=u/2+1/2sin u\ cos u+K`

10. `intcos^3u\ du` `=sin u-1/3sin^3u+K`

11. `intcos^(n)u\ du` `=1/ncos^(n-1)u\ sin u` `+(n-1)/nintcos^(n-2)u\ du`

12. `inttan^(n)u\ du` `=(tan^-1u)/(n-1)-inttan^(n-2)u\ du`

13. `int(du)/(u^2-a^2)` `=1/(2a) ln\ |(u-a)/(u+a)|+K`

14. `int(du)/(sqrt(u^2+-a^2))` `=ln\ |u+sqrt(u^2+-a^2)|+K`

15. `intt\ sin nt\ dt` `=1/(n^2)(sin nt-nt\ cos nt)+K`

16. `intt\ cos nt\ dt` `=1/(n^2)(cos nt+nt\ sin nt)+K`

17. `inte^(au)\ sin bu\ du` `=(e^(au)(a\ sin bu-b\ cos bu))/(a^2+b^2)+K`

18. `inte^(au)cos bu\ du` `=(e^(au)(a\ cos bu+b\ sin bu))/(a^2+b^2)+K`

19. `intu^(au)du` `=e^(au)(a^2u^2-2au+2)/(a^3)+K`

20. `intt^2\ sin nt\ dt` `=1/n^3(-n^2t^2cos nt` `{:+2\ cos nt+2nt\ sin nt)+K`

21. `intt^2cos ntdt` `=1/(n^3)(n^2t^2\ sin nt-2\ sin nt` `{:+2nt\ cos nt)+K`

Problem Solver

AI Math Calculator Reviews

This tool combines the power of mathematical computation engine that excels at solving mathematical formulas with the power of GPT large language models to parse and generate natural language. This creates math problem solver thats more accurate than ChatGPT, more flexible than a calculator, and faster answers than a human tutor. Learn More.

Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.