Integral

An indefinite integral is the inverse of a derivative. According to the fundamental theorem of calculus, if the integral of a function f(x) equals F(x) + K, then the derivative of F(x) equals f(x). This is true for any numerical value of the constant K, and so the integral is called indefinite.

The inverse relationship between derivative and integral has two very important consequences. First, in many practical applications, the functional relationship between two quantities is unknown, and not easily measured. However, the rate at which one of these quantities changes with respect to the other is known, or easily measured (for instance the previous example of the work done on a piston). Knowing the rate at which one quantity changes with respect to another means that the derivative of the one with respect to the other is known (since that is just the definition of derivative). Thus, the underlying functional relationship between two quantities can be found by taking the integral of the derivative. The second important consequence arises in evaluating definite integrals. Many times it is exceedingly difficult, if not impossible, to find the value of the integral. However, a relatively easy method, by comparison, is to find a function whose derivative is the function to be integrated, which is then the integral.