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The constant of integration is an unknown constant that must be taken into account when taking an indefinite integral. Since the derivative of any constant is 0, any constants will be "lost" when differentiating. The constant of integration is usually represented with C , or, in the case of a differential equation where there are multiple constants, C_1,C_2,\ldots

In integral calculus, the constant of integration is usually added to the end of a function; for example:

\int f(x)dx=F(x)+C

However, in differential equations, the constant of integration is often used in other operations on the final function (the most common is multiplication). For example:

\frac{dy}{dx}=xy
y=\pm Ce^\frac{x^2}{2}

There can also be multiple constants of integration.

\frac{d^2y}{dx^2}-3\frac{dy}{dx}-4y=0
y=C_1e^{4x}+C_2e^{-x}

In the case of partial derivatives, the constant of integration is not a constant, but a function of all the independent variables save for the one being integrated (in this case, we'll say this is x). This is necessary because any independent variables that are not x will be treated as constants, so any function that does not have an x in them will be lost. For example,

\frac{\part z}{\part x}=x^2+3y
\int\dfrac{\part z}{\part x}dx=\int(x^2+3y)dx=\frac{x^3}{3}+3yx+f(y)

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