Integration 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 ca 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) + ca (n ≠ -1)
- ∫x-1 dx = ln(|x|) + ca
- ∫ln(x) dx = x ln(x) - x + ca
- ∫ex dx = ex + ca
- ∫ax dx = ax/ln(a) + ca
- ∫(1+x2)-1 dx = arctan(x) + ca
- ∫(1-x2)-1/2 dx = arcsin(x) + ca
- ∫cos(x) dx = sin(x) + ca
- ∫sin(x) dx = -cos(x) + ca
- ∫tan(x) dx = -ln|cos(x)| + ca
- ∫csc(x) dx = -ln|csc(x)+cot(x)| + ca
- ∫sec(x) dx = ln|sec(x)+tan(x)| + ca
- ∫cot(x) dx = ln|sin(x)| + ca
- ∫sec2(x) dx = tan(x) + ca
- ∫csc2(x) dx = -cot(x) + ca
- ∫sin2(x) dx = x/2-(sin(2x))/4 + ca
- ∫cos2(x) dx = x/2+(sin(2x))/4 + ca
- ∫sinh(x) dx = cosh(x) + ca
- ∫cosh(x) dx = sinh(x) + ca
- ∫tanh(x) dx = ln(cosh(x)) + ca
- ∫csch(x) dx = ln|tanh(x/2)| + ca
- ∫sech(x) dx = arctan(sinh(x)) + ca
- ∫coth(x) dx = ln|sinh(x)| + ca