Editing Constant of integration (section)

==Significance==

The inclusion of the constant of integration is necessitated in some, but not all circumstances. For instance, when evaluating [[definite integral]]s using the [[fundamental theorem of calculus]], the constant of integration can be ignored as it will always cancel with itself.

However, different methods of computation of indefinite integrals can result in multiple resulting antiderivatives, each implicitly containing different constants of integration, and no particular option may be considered simplest. For example, <math>2\sin(x)\cos(x)</math> can be integrated in at least three different ways.

<math display="block">\begin{alignat}{4}
\int 2\sin(x)\cos(x)\,dx =&&  \sin^2(x) + C =&& -\cos^2(x) + 1 + C =&& -\frac 1 2 \cos(2x) + \frac 1 2 + C\\
\int 2\sin(x)\cos(x)\,dx =&& -\cos^2(x) + C =&&  \sin^2(x) - 1 + C =&& -\frac 1 2 \cos(2x) - \frac 1 2 + C\\
\int 2\sin(x)\cos(x)\,dx =&& -\frac 1 2 \cos(2x) + C =&& \sin^2(x) + C =&& -\cos^2(x) + C \\
\end{alignat}</math>Additionally, omission of the constant, or setting it to zero, may make it prohibitive to deal with a number of problems, such as those with [[Initial value problem|initial value conditions]]. A general solution containing the arbitrary constant is often necessary to identify the correct particular solution. For example, to obtain the antiderivative of <math>\cos(x)</math> that has the value 400 at ''x'' = π, then only one value of <math>C</math> will work (in this case <math>C = 400</math>).

The constant of integration also implicitly or explicitly appears in the language of [[differential equations]]. Almost all differential equations will have many solutions, and each constant represents the unique solution of a well-posed initial value problem.

An additional justification comes from [[abstract algebra]]. The space of all (suitable) real-valued functions on the [[real number]]s is a [[vector space]], and the [[differential operator]] <math display="inline">\frac{d}{dx}</math> is a [[linear operator]]. The operator <math display="inline">\frac{d}{dx}</math> maps a function to zero if and only if that function is constant.  Consequently, the [[kernel (algebra)|kernel]] of <math display="inline">\frac{d}{dx}</math> is the space of all constant functions.  The process of indefinite integration amounts to finding a pre-image of a given function. There is no canonical pre-image for a given function, but the set of all such pre-images forms a [[coset]]. Choosing a constant is the same as choosing an element of the coset. In this context, solving an [[initial value problem]] is interpreted as lying in the [[hyperplane]] given by the [[initial condition]]s.