Integration table

October 2, 2017 Mouctar Calculus 0

Integration table

In the previous chapters, we have used an extensive amount of formulas for the indefinite integration.

While we have not started the definite integrals calculation, it is handy to have some of the intagration formulas in a given place for easy reference.

It is obvious that we are now able to demonstrate how to obtain the majority of these formulas by using the intergration techniques learned.

Table of some integrals

01. \int x^{n}\; dx=\frac{1}{n+1}x^{n+1}+C
02. \int \frac{dx}{x}=\ln x+C
03. \int a^{x} \;dx=\frac{a^{x}}{\ln a}+C
04. \int e^{x} \;dx=e^{x}+C
05. \int \sin x \;dx=-\cos x+C
06. \int \sin ax \;dx=-\frac{1}{a}\cos ax+C
07. \int \cos x \;dx=\sin x+C
08. \int \cos ax \;dx=\frac{1}{a}\sin ax+C
09. \int \tan x \;dx=-\ln |\cos x|+C
10. \int \tan x \;dx=\ln |\sec x|+C
11. \int \cot x \;dx=\ln |\sin x|+C
12. \int \cot x \;dx=-\ln |\csc x|+C
13. \int \sec x \;dx=\ln |\sec x + \tan x|+C
14. \int \csc x \;dx=\ln |\csc x - \cot x|+C
15. \int \sec^{2} x \;dx=\tan x+C
16. \int \csc^{2} x \;dx=-\cot x+C
17. \int \sec x \tan x \;dx=\sec x+C
18. \int \csc x \cot x \;dx=-\csc x+C
19. \int \sinh x \;dx=\cosh x+C
20. \int \cosh x \;dx=\sinh x+C
21. \int \tanh x \;dx=\ln \cosh x+C
22. \int \coth x \;dx=\ln |\sinh x|+C
23. \int sech^{2} x \;dx=\tanh x+C
24. \int csch^{2} x \;dx=-\coth x+C
25. \int sech x \tanh x \;dx=-sech x+C
26. \int csch x \coth x \;dx=-csch x+C
27. \int \frac{dx}{a^{2}+x^{2}}=\frac{1}{a} \arctan \frac{x}{a}+C
28. \int \frac{dx}{\sqrt{a^{2}-x^{2}}}=\arcsin \frac{x}{a}+C
29. \int \frac{dx}{\sqrt{x^{2}-a^{2}}}=\ln |x + \sqrt{x^{2}-a^{2}} |+C
30. \int \frac{dx}{\sqrt{x^{2}-a^{2}}}=\cosh^{-1} \frac{x}{a}+C
31. \int \frac{dx}{\sqrt{x^{2}+a^{2}}}=\ln |x + \sqrt{x^{2}+a^{2}} |+C
32. \int \frac{dx}{\sqrt{x^{2}+a^{2}}}=\sinh^{-1} \frac{x}{a}+C
33. \int \frac{dx}{x\sqrt{x^{2}-a^{2}}}=\frac{1}{a}\sec^{-1} \frac{x}{a}+C
34. \int \frac{dx}{x^{2}-a^{2}}=\frac{1}{2a} \ln | \frac{x-a}{x+a} |+C
35. \int \frac{dx}{x^{2}-a^{2}}=-\frac{1}{a} \coth^{-1}\frac{x}{a} |+C
36. \int \frac{dx}{a^{2}-x^{2}}=\frac{1}{2a} \ln | \frac{a+x}{a-x} |+C
37. \int \frac{dx}{a^{2}-x^{2}}=\frac{1}{a} \tanh^{-1}\frac{x}{a} |+C
38. \int u \; dv=uv- \int v \; du
39. \int \sin^{n} u \; du=-\frac{1}{n}\sin^{n-1} u \cos u+ \frac{n-1}{n} \int \sin^{n-2} u\; du
40. \int \cos^{n} u \; du=\frac{1}{n}\cos^{n-1} u \sin u+ \frac{n-1}{n} \int \cos^{n-2} u\; du
41. \int \tan^{n} u \; du=\frac{1}{n-1}\tan^{n-1} u-\int \tan^{n-2} u\; du
42. \int \cot^{n} u \; du=\frac{-1}{n-1}\cot^{n-1} u-\int \cot^{n-2} u\; du
43. \int \sec^{n} u \; du=\frac{1}{n-1}\sec^{n-2} u \tan u+ \frac{n-2}{n-1} \int \sec^{n-2} u\; du
44. \int \csc^{n} u \; du=\frac{-1}{n-1}\csc^{n-2} u \cot u+ \frac{n-2}{n-1} \int \csc^{n-2} u\; du
45. \int \sin^{n} u \cos^{m} u \; du=-\frac{\sin^{n-1} u \cos^{m+1} u}{n+m}+ \frac{n-1}{n+m} \int \sin^{n-2} u \cos^{m} u\; du
46. \int \sin^{n} u \cos^{m} u \; du=\frac{\sin^{n+1} u \cos^{m-1} u}{n+m}+ \frac{m-1}{n+m} \int \sin^{n} u \cos^{m-2} u\; du
47. \int \frac{du}{u^{2}\sqrt{u^{2}\pm a^{2}}}=\mp \frac{\sqrt{u^{2}\pm a^{2}}}{a^{2}u}
48. \int \frac{du}{u^{2}\sqrt{a^{2}-u^{2}}}=-\frac{\sqrt{a^{2}-u^{2}}}{a^{2}u}
49. \int \frac{du}{\sqrt{u^{2}+a^{2}}}=\ln |u + \sqrt{u^{2}+a^{2}} |+C
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