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# Table of Common Integrals

### Leibniz's table of derivatives and 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