Table of Integrals

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Integration and finding antiderivatives is one of the two basic operations in calculus and since it, unlike differentiation, is non-trivial, tables of known integrals are often useful. Here is the beginning of such a table.

We use C for an arbitrary constant that can only be determined if something about the value of the integral at some point is known.

∫xn dx = xn+1/(n+1) + C    (n ≠ -1)
∫x-1 dx = ln(|x|) + C

∫ln(x) dx = x ln(x) - x + C

∫ex dx = ex + C
∫ax dx = ax/ln(a) + C

∫(1+x2)-1 dx = arctan(x) + C
-∫(1+x2)-1 dx = arccot(x) + C
∫(1-x2)-1/2 dx = arcsin(x) + C
-∫(1-x2)-1/2 dx = arccos(x) + C
∫x(x2-1)-1/2 dx = arcsec(x) + C
-∫x(x2-1)-1/2 dx = arccsc(x) + C

∫cos(x) dx = sin(x) + C
∫sin(x) dx = -cos(x) + C
∫tan(x) dx = -ln|cos(x)| + C
∫csc(x) dx = -ln|csc(x)+cot(x)| + C
∫sec(x) dx = ln|sec(x)+tan(x)| + C
∫cot(x) dx = ln|sin(x)| + C

∫sec2(x) dx = tan(x) + C
∫csc2(x) dx = -cot(x) + C
∫sin2(x) dx = x/2-(sin(2x))/4 + C
∫cos2(x) dx = x/2+(sin(2x))/4 + C

∫sinh(x) dx = cosh(x) + C
∫cosh(x) dx = sinh(x) + C
∫tanh(x) dx = ln(cosh(x)) + C
∫csch(x) dx = ln|tanh(x/2)| + C
∫sech(x) dx = arctan(sinh(x)) + C
∫coth(x) dx = ln|sinh(x)| + C