Constant of integration

In <a href="/facts/Calculus/MEmUTYoz">calculus</a>, the constant of integration, often denoted by 
 
 
 
 C
 
 
 {\displaystyle C}
 
 (or 
 
 
 
 c
 
 
 {\displaystyle c}
 
), is a <a href="/facts/Constant_term/KRFu0aum">constant term</a> added to an <a href="/facts/Antiderivative/5qDsudIv">antiderivative</a> of a function 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 to indicate that the <a href="/facts/Indefinite_integral/5qDsudIv">indefinite integral</a> of 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 (i.e., the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of all antiderivatives of 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
), on a <a href="/facts/Connected_set/5CUTxfoA">connected domain</a>, is only defined <a href="/facts/Up_to/ydz4ztzX">up to</a> an additive constant. This constant expresses an ambiguity inherent in the construction of antiderivatives.
More specifically, if a function 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 is defined on an <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a>, and 
 
 
 
 F
 (
 x
 )
 
 
 {\displaystyle F(x)}
 
 is an antiderivative of 
 
 
 
 f
 (
 x
 )
 ,
 
 
 {\displaystyle f(x),}
 
 then the set of all antiderivatives of 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 is given by the functions 
 
 
 
 F
 (
 x
 )
 +
 C
 ,
 
 
 {\displaystyle F(x)+C,}
 
 where 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is an arbitrary constant (meaning that any value of 
 
 
 
 C
 
 
 {\displaystyle C}
 
 would make 
 
 
 
 F
 (
 x
 )
 +
 C
 
 
 {\displaystyle F(x)+C}
 
 a valid antiderivative). For that reason, the indefinite integral is often written as 
 
 
 
 ∫
 f
 (
 x
 )
 
 d
 x
 =
 F
 (
 x
 )
 +
 C
 ,
 
 
 {\textstyle \int f(x)\,dx=F(x)+C,}
 
 although the constant of integration might be sometimes omitted in <a href="/facts/Lists_of_integrals/FqW2wdwr">lists of integrals</a> for simplicity.

Constant of integration open-in-new

Constant of integration