# Definite Integral

Consider an expression $F\left( x \right)$ such that

Integrating both sides with respect to $x$, we have

Putting the value $x = a$ in equation (i), we have

Similarly, putting the value $x = b$ in equation (i), we have

Subtracting $F\left( b \right)$ from $F\left( a \right)$, we have

We can write the above expression as

Since ${x^4}$ is the anti-derivative $4{x^3}$, so $\int {4{x^3}dx = {x^4}}$, where we have ignored $c$ as it is cancelled out in (ii). Making this substitution in (ii), we have

We conclude that if $F\left( x \right)$ is the anti-derivative of $f\left( x \right)$, i.e.

Then it can be written as

Thus, we have

The integral $\int\limits_a^b {f\left( x \right)dx}$ is known as a definite integral.