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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) `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)`