The Definite Integral

Consider an expression $$F\left( x \right)$$ such that
\[\frac{d}{{dx}}\left[ {f\left( x \right)} \right] = 4{x^3}\]

Integrating both sides with respect to $$x$$, we have
\[\begin{gathered} \int {d\left[ {f\left( x \right)} \right]} = \int {4{x^3}dx} \\ \Rightarrow F\left( x \right) = \frac{{4{x^4}}}{4} + c \\ \Rightarrow F\left( x \right) = {x^4} + c\,\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered} \]

Putting the value $$x = a$$ in equation (i), we have
\[F\left( a \right) = {a^4} + c\]

Similarly, putting the value $$x = b$$ in equation (i), we have
\[F\left( b \right) = {b^4} + c\]

Subtracting $$F\left( b \right)$$ from $$F\left( a \right)$$, we have
\[F\left( b \right) – F\left( a \right) = \left( {{b^4} + c} \right) – \left( {{a^4} + c} \right) = {b^4} – {a^4}\]

We can write the above expression as
\[F\left( b \right) – F\left( a \right) = \left| {{x^4}} \right|_a^b\,\,\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right)\]

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
\[F\left( b \right) – F\left( a \right) = \left| {\int {4{x^3}dx} } \right|_a^b\]
\[F\left( b \right) – F\left( a \right) = \int\limits_a^b {4{x^3}dx} \]

We conclude that if $$F\left( x \right)$$ is the anti-derivative of $$f\left( x \right)$$, i.e.
\[\frac{d}{{dx}}\left[ {F\left( x \right)} \right] = f\left( x \right)\]

Then it can be written as
\[\int\limits_a^b {f\left( x \right)dx = F\left( b \right) – F\left( a \right)} \]

Thus, we have
\[\frac{d}{{dx}}\left[ {f\left( x \right)} \right] = f\left( x \right) \Rightarrow \int\limits_a^b {f\left( x \right)dx = F\left( b \right) – F\left( a \right)} \]

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