JEE Main & Advanced Mathematics Definite Integration Definition

Let \[\varphi (x)\] be the primitive or anti-derivative of a function \[f(x)\] defined on \[[a,\,\,b]\] i.e., \[\frac{d}{dx}[\varphi (x)]=f(x)\]. Then the definite integral of  \[f(x)\] over \[[a,\,\,b]\] is denoted by \[\int_{a}^{b}{f(x)dx}\] and is defined as \[[\varphi (b)-\varphi (a)]\] i.e., \[\int_{a}^{b}{f(x)dx=\varphi (b)-\varphi (a)}\]. This is also called Newton Leibnitz formula.

The numbers \[a\] and \[b\] are called the limits of integration, \['a'\] is called the lower limit and \['b'\] the upper limit. The interval \[[a,\,\,b]\]is called the interval of integration. The interval \[[a,\,\,b]\] is also known as range of integration. Every definite integral has a unique value.