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Table of Common Integrals

Leibniz's table of integrals

Leibniz table of integrals
A simple table of integrals from the Gottfried Leibniz archive. [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`

Leibniz's original integration notation

Leibniz original integration notation
Gottfried Leibniz's original notation for integrals included an overbar, similar to what we use with square roots today. [Image source]

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`

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