Skip to main content

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`

top

Search IntMath, blog and Forum

Search IntMath

Online Algebra Solver

This algebra solver can solve a wide range of math problems.

Calculus Lessons on DVD

Math videos by MathTutorDVD.com

Easy to understand calculus lessons on DVD. See samples before you commit.

More info: Calculus videos

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!


See the Interactive Mathematics spam guarantee.