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. |
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. |
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. |
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) + ca (n ≠ -1) :∫x-1 dx = ln(|x|) + ca |
:∫xn dx = xn+1/(n+1) + C (n ≠ -1) :∫x-1 dx = ln(|x|) + C |
:∫ln(x) dx = x ln(x) - x + ca |
:∫ln(x) dx = x ln(x) - x + C |
:∫ex dx = ex + ca :∫ax dx = ax/ln(a) + ca |
:∫ex dx = ex + C :∫ax dx = ax/ln(a) + C |
:∫(1+x2)-1 dx = arctan(x) + ca :-∫(1+x2)-1 dx = arccot(x) + ca :∫(1-x2)-1/2 dx = arcsin(x) + ca :-∫(1-x2)-1/2 dx = arccos(x) + ca :∫x(x2-1)-1/2 dx = arcsec(x) + ca :-∫x(x2-1)-1/2 dx = arccsc(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 |
:∫(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 |
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.