Math Is Fun Forum

Shelled

Member

Registered: 2014-04-15

Posts: 44

Constant of Integration

Why do we need to add a constant, C, when we're solving an indefinite integral?

ShivamS

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Registered: 2011-02-07

Posts: 3,648

Re: Constant of Integration

An indefinite integral or antiderivative has no specified limits for the integration. For application to specific problems, boundary conditions must be applied to the result in order to arrive at a specific value for the integral. The uncertainty in the value of the indefinite integral is expressed in the form of a constant of integration which is not defined by the integration process.

zetafunc

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Registered: 2014-05-21

Posts: 2,438

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Re: Constant of Integration

I've split my post into two parts. Feel free to read the more elaborate explanation if you'd like to learn a little more about indefinite integrals.

Quick reason: The constant is needed because it is still technically a possibility for the function. For instance, consider the integral:

.

Clearly, one solution is x². Another is x² + 2. Another is x² - 5. They all differentiate to form 2x. Hence, we add a 'constant of integration' to account for this.

More elaborate explanation:

To add to ShivamS' answer, a slightly more precise way of defining an indefinite integral is as follows. Let f be locally Riemann integrable over I. Then, an indefinite integral of f is a function F: I -> R defined by

for some a ∈ I. From the domain-splitting property for integrals, it follows that two indefinite integrals differ by a constant. Further, it follows that:

(i) F is continuous on I;
(ii) F is differentiable at each interior point c ∈ I at which f is continuous, and satisfies F'(c) = f(c);
(iii) If f is continuous on I, then F is clearly a primitive of f.

Such a primitive is not unique, because one can always add a constant to F.

(As an aside, you might like to know that (ii) allows one to differentiate integrals!)

Last edited by zetafunc (2014-05-23 20:09:47)