Arbitrary constant of integration

In calculus, the indefinite integral of a given function (i.e. the set of all antiderivatives of the function) is often written with a constant, the constant of integration. The reason is that if a function f is defined on an interval and F is an antiderivative of f, then the set of all antiderivatives of f is given by the functions F(x) + c, with c an arbitrary constant.

For example, the formula

gives a compact description of all antiderivaties of the function f(x) = cos(x); C here serves as the constant of integration.

Occasionally, it is necessary to find a particular antiderivative F of f with a given condition, such as F(0) = 0. This can be done by first finding the indefinite integral, and then solving for the particular value of the constant of integration determined by the condition.

In indefinite integrals, the constant of integration is always an additive one. When solving certain ordinary differential equations, for instance with the method of separation of variables, one has to keep track of the integration constants because they determine the set of solutions to the differential equation, and not all of them remain additive.

While it may seem that there is a "simplest integral", whose constant of integration is C = 0, this is not so. For example, consider the integral of -2cos(x)sin(x): if I(x) = cos²(x), and J(x) = -sin²(x), then note that I(x) = J(x) + 1; so the set of functions of the form {J(x) + C} is equal to the set of functions of the form {I(x) + C}. Thus, the choice of whether I(x) or J(x) is "simplest" is entirely arbitrary.

In the language of abstract algebra, the presence of C shows that indefinite integrals are actually cosets, with respect to the kernel of the differentiation map that is being inverted.