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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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