# Definite Integral

The definite integral of *f*(*x*) is the difference between two values of the integral of *f*(*x*) for two distinct values of the variable *x*. If the integral of *f*(*x*) *dx* = *F*(*x*) + *C*, the definite integral is denoted by the symbol

The quantity *F*(*b*) - *F*(*a*) is called the definite integral of *f*(*x*) between the limits *a* and *b* or simply the definite integral from *a* to *b*. It is called the definite integral because the result involves neither *x* nor the constant *C* and therefore has a definite value. The numbers *a* and *b* are called the limits of integration, *a* being the lower limit and *b* the upper limit.

### General Properties of Definite Integral

- The sign of the integral changes if the limits are interchanged.

$$\displaystyle \int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$$

- The interval of integration may be broken up into any number of sub-intervals, and integrate over each interval separately.

$$\displaystyle \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$$

- The definite integral of a given integrand is independent of the variable of integration. Hence, it makes no difference what letter is used for the variable of integration.

$$\displaystyle \int_a^b f(x) \, dx = \int_a^b f(z) \, dz$$