Arbitrary constant of integration - Where does the constant come from?

In calculus, the indefinite integral of a given function (i.e. the set of all antiderivatives of the function) is always written with a constant, the constant of integration. This constant expresses an ambiguity inherent in the construction of antiderivatives. 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 funct ...

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• Arbitrary constant of integration - Where does the constant come from?
• Arbitrary constant of integration - Why is the constant necessary?
• Arbitrary constant of integration - Why is a constant the only difference between two antiderivatives?

Read more here: » Arbitrary constant of integration: Encyclopedia - Arbitrary constant of integration

The derivative of any constant function is zero. Once one has found one antiderivative F, adding or subtracting a constant C will give us another antiderivative, because (F + C)' = F' + C' = F' . The constant is a way of expressing that every function has an infinite number of different antiderivatives. For example, suppose one wants to find antiderivatives of cos(x). One such antiderivative is sin(x). Another one is sin(x)+1. A third is sin(x)-π. Each ...

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Arbitrary constant of integration, Arbitrary constant of integration - Where does the constant come from?, Arbitrary constant of integration - Why is the constant necessary?, Arbitrary constant of integration - Why is a constant the only difference between two antiderivatives?

Read more here: » Arbitrary constant of integration: Encyclopedia II - Arbitrary constant of integration - Where does the constant come from?

At first glance it may seem that the constant is unnecessary, since it can be set to zero. Even better, when evaluating definite integrals using the Fundamental theorem of calculus, the constant will always cancel. However, trying to set the constant equal to zero doesn't always make sense. For example, 2sin(x)cos(x) can be integrated in two different ways: So setting C to zero can still leave a con ...

See also:

Arbitrary constant of integration, Arbitrary constant of integration - Where does the constant come from?, Arbitrary constant of integration - Why is the constant necessary?, Arbitrary constant of integration - Why is a constant the only difference between two antiderivatives?

Read more here: » Arbitrary constant of integration: Encyclopedia II - Arbitrary constant of integration - Why is the constant necessary?