What exactly does the constant c stand for in an indefinite integral?

Question

When I solve indefinite integrals, I tag on the +c to the answer. Here is an example:

Integral (x^2 dx) = (1/3)X^3 + C

But what is the "C" and what does it mean? Can you also explain it for me for multi-variable integrals too?

Wow! Thanks to everyone, I understand now. I wish I could give you all a best answer.

Answer 1

If one differentiates a function that includes a constant, such as f(x) = sin(x) + C, the result is f '(x) = cos(x) + 0, because the differential of the constant is zero. When one integrates, one is reversing the process, 'taking the anti-derivative', and it is necessary to replace the constant. In an indefinite integral, it is unknown, and one simply 'adds C' as you have done. There is no way to know the specific value of the constant without being given a 'boundary condition,' a value of f(x) at some specific value of x.

If one has a multivariable derivative, such as ∂²F/∂x∂y = 3x²y, and one integrates with respect to x to obtain ∫ 3x²y dx = x³y + C(y), the 'constant' C in this case is a function of y, because if F(x,y) = x³y + y², say, then∂F/∂x = 3x²y, and the y²-term 'disappears,' just as the constant 'disappeared' in the first example.

Hope this helps.

Answer 2

The C is a constant of integration. Think of it like this.
(1/3)x^3 is a solution to the Integral (x^2 dx) because it is the antiderivative, but (1/3)x^3 + C is also a solution because the derivative of [(1/3)x^3 + C] = x^2, since the d/dx{C} = 0. So there is always a constant associated with the antiderivative.

Answer 3

C is an arbitrary constant known as the constant of integration.

This constant expresses an ambiguity inherent in the construction of antiderivatives.

Since the derivative of a constant is zero, any constant may be added to an indefinite integral (i.e., antiderivative) and will still correspond to the same integral.

This means that, depending on the form used for the integrand, antiderivatives F_1 and F_2 can be obtained that differ by a constant.

I guess you could simplify it by saying it is a place-holder for a value that was zero'd out during initial derivation and cannot be determined satisfactorily without more information on the original equation.

Answer 4

It can mean anything. Remember that integrating is essentially reversing a derivative, and then recall than when you take derivatives, constant terms disappear.

When you try to take an integral, there is no way to magically make those constants reappear. Instead, we throw the C on there to say "we know there might be something here, but the problem doesn't contain enough information to know what it was".

If you have a set of initial conditions, you can solve your answer for C to get the actual number back.

Answer 5

It's a value that is essentially "lost" in the differentiation, and must be restored outside of the integration.

Consider y' = 3

That defines a line with slope=3. There are infinitely many such lines:

y = 3x, y = 3x + 1, y = 3x + 2, y = 3x + 99999 etc.

When integrating to get y from y', which line do we choose? We write y = 3x + C and leave it at that, since that defines all solutions to y' = 3.

If there is some additional information (for example, the line passes through the x-axis at x=4), then we can use that information to identify the value for C that matches the additional constraint.

Answer 6

An indefinite integral defines a family of curves separated by an arbitrary constant C. It can have any value and is only defined when you are solving for a specific set of conditions.
The same thing applies with multiple variables.

Answer 7

It stands for any arbitrary constant. Basically what it's saying is that there are an infinite number of functions whose derivative equals x^2. For example:

d/dx ((1/3)X^3 + 0) = x^2
d/dx ((1/3)X^3 + 17) = x^2
d/dx ((1/3)X^3 + 6.22) = x^2
d/dx ((1/3)X^3 + 94.9) = x^2
:
etc. So when you find the ANTI-derivative of x^2, the answer is not a unique function, it's an infinite family of functions. They express that by saying:

anti-derivative of x^2 = (1/3)X^3 + (any constant)

Answer 8

indefinate integral is nothing , just a search of a function who's differentiation is that ..
as f(x)= x^2+2
then f'(x)=2x
and integrate it again it must be the question ..
let f'(x)=g(x)
so when integrate g(x)
it is x^2
bt our question is x^2+2...!!
and if it is x^2+100 then answer is also x^2
so mathematicians come to the result that there may be any constant and that is represent by 'c'
actully indefinite integral has no significance geometrically ...
it is just taught to the student to make out the definite integral...

Answer 9

C is called a constant of integration. Its value depends on the initial conditions of the problem.

Answer 10

0.25e^(4x) - 4*ln(x) + C