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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) ∫
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)