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Table of Integrals - Basic Forms and Common Integrals

The integrals below are very common and are used in a great many calculus problems.  We include basic polynomial rules, integrals of the exponential function, integrals of basic trig functions, and more.

1) `int  a  dx = ax`

2) `int  x^n  dx= 1/(n+1) x^(n+1)`

3) `int  1/x  dx = ln|x|`

4) `int  e^x  dx = e^x`

5) `int  a^x  dx = int e^(x ln a) dx = (e^(x ln a))/ln a = a^x/ln a, a > 0, a!=1`


6) `int  sin x  dx = - cos x`

7) `int  cos x  dx = sin x`

8) `int  tan x  dx = ln sec x = -ln cos x`

9) `int  cot x  dx = ln sin x`

10) `int  sec x  dx = ln (sec x + tan x) = ln [tan (x/2 + pi/4)]`

11) `int  csc x  dx = ln (csc x - cot x) = ln [tan (x/2)]`

12) `int  sec^2 x  dx = tan x`

13) `int  csc^2 x  dx = - cot x`

14) `int  tan^2 x  dx = tan x - x`

15) `int  cot^2 x  dx = - cot x - x`

16) `int  sin^2 x  dx = x/2 - (sin 2x)/4 = 1/2 (x - sin x cos x)`

17) `int  cos^2 x  dx = x/2 + (sin 2x)/4 = 1/2 (x + sin x cos x)`

18) `int  sec x tan x  dx = sec x`

19) `int  csc x cot x  dx = - csc x`


20) `int  sinh x  dx = cosh x`

21) `int  cosh x  dx = sinh x`

22) `int  tanh x  dx = ln (cosh x)`

23) `int  coth x  dx = ln (sinh x)`

24) `int  sech x  dx = sin^-1 (tanh x) = 2 tan^-1 e^x`

25) `int  csch x  dx = ln [tanh (x/2)] = - coth^-1 e^x`

26) `int  sech^2 x  dx = tanh x`

27) `int  csch^2 x  dx = - coth x`

28) `int  tanh^2 x  dx = x - tanh x`

29) `int  coth^2 x  dx = x - coth x`

30) `int  sinh^2 x  dx = (sinh 2x)/4 - x/2 = 1/2 (sinh x cosh x - x)`

31) `int  cosh^2 x  dx = (sinh 2x)/4 + x/2 = 1/2 (sinh x cosh x + x)`

32) `int  sech x tanh x  dx = - sech x`

33) `int  csch x coth x  dx = - csch x`


34) `int  1/(x^2 + a^2)  dx = 1/a tan^-1 (x/a)`

35) `int  1/(x^2 - a^2)  dx = 1/(2a) ln((x-a)/(x+a)) = - 1/a coth^-1 (x/a), x^2 > a^2`

36) `int  1/(a^2 - x^2)  dx = 1/(2a) ln((a+x)/(a-x)) = 1/a tanh^-1 (x/a), x^2 < a^2`

37) `int  1/sqrt(a^2 - x^2)  dx = sin^-1 (x/a)`

38) `int  1/sqrt(x^2 + a^2)  dx = ln(x + sqrt(x^2 + a^2)) = sinh^-1 (x/a)`

39) `int  1/sqrt(x^2 - a^2)  dx = ln(x + sqrt(x^2 - a^2))`

40) `int  1/(x sqrt(x^2 - a^2))  dx = 1/a sec^-1 |x/a|`

41) `int  1/(x sqrt(x^2 + a^2))  dx = - 1/a ln((a+sqrt(x^2 + a^2))/x)`

42) `int  1/(x sqrt(a^2 - u^2))  dx = - 1/a ln((a+sqrt(a^2 - u^2))/x)`


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