Integration of Functions | CUET PG Economics | Notes

Meaning and Concept of Integration

Integration is the mathematical process of finding a function whose derivative is a given function. It is essentially the reverse operation of differentiation, and for this reason integration is often referred to as antidifferentiation. While differentiation helps us determine rates of change — such as marginal cost from total cost — integration allows us to move in the opposite direction: recovering the total function from information about its rate of change. In economics, this is extremely useful because many relationships are naturally expressed in marginal terms, and integration allows economists to derive total functions such as total cost, total revenue, and total utility from their respective marginal functions.

If \(F(x)\) is a function such that \(F'(x) = f(x)\), then \(F(x)\) is called the antiderivative or integral of \(f(x)\). This relationship is written as:

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

Here, (C) is called the constant of integration, which arises because the derivative of any constant is zero, meaning infinitely many functions can have the same derivative. The symbol \(\int\) denotes the integral sign, \(f(x)\) is called the integrand, and \(dx\) indicates that the integration is being performed with respect to the variable (x).