The indefinite integral is the inverse of the derivative, that is, the integral of the derivative of a function is the original function. From the definition of derivative, f'(x) = df(x)/dx, we find f'(x)dx = df(x). To obtain the original function from this second equation, we "integrate" both sides, and write, ∫ f'(x)dx = ∫df(x) = f(x) + C. The integral sign, ∫ , is intended to symbolize the summing, integrating, or putting together of the infinitesimal pieces df(x) to obtain the original function f(x). The constant, C, arises because functions that differ only by a constant are "parallel" to one another, and so have the same derivative (or slope) at each value of x. Defined in this manner, integrating amounts to guessing original functions based on prior knowledge of their derivatives. For example, if f(x) = x2 then f'(x) =2x. Thus, if asked to integrate the function g(x) = 2x, it is apparent that ∫2xdx = x2 + C. In order to determine the value of C it is necessary to have an additional piece of information. Such information is referred to as an initial condition or a boundary condition, and is sufficient to determine which of the parallel curves is the desired one.
The primary application of indefinite integrals is in the solution of differential equations. A differential equation is any equation that contains at least one derived. The equation f'(x) = ax2+bx+c is an example of a differential equation. Many natural relationships are described by differential equations. For instance, heat conduction is related to the derivative of the temperature with respect to distance; the velocity of a fluid flowing through a pipe is related to the derivative of the pressure with respect to length of pipe; and the force on any massive body is related to the derivative of its momentum with respect to time.