# The Definite Integral of a Linear Function

In this tutorial we shall find an example of a definite integral of a linear function from limits 1 to 2.

The integration is of the form
$I = \int\limits_1^2 {\left( {4x + 1} \right)dx}$

We use the basic rule of definite integral $\int\limits_a^b {f\left( x \right)dx = \left| {F\left( x \right)} \right|_a^b} = \left[ {F\left( b \right) – F\left( a \right)} \right]$, and we have
$\begin{gathered} I = 4\int\limits_1^2 {xdx} + \int\limits_1^2 {1dx} \\ \Rightarrow \int\limits_1^2 {\left( {4x + 1} \right)dx} = 4\left| {\frac{{{x^2}}}{2}} \right|_1^2 + \left| x \right|_1^2 \\ \Rightarrow \int\limits_1^2 {\left( {4x + 1} \right)dx} = 2\left| {{x^2}} \right|_1^2 + \left| x \right|_1^2 \\ \Rightarrow \int\limits_1^2 {\left( {4x + 1} \right)dx} = 2\left[ {{{\left( 2 \right)}^2} – {{\left( 1 \right)}^2}} \right] + \left[ {2 – 1} \right] \\ \Rightarrow \int\limits_1^2 {\left( {4x + 1} \right)dx} = 2\left[ {4 – 1} \right] + \left[ {2 – 1} \right] \\ \Rightarrow \int\limits_1^2 {\left( {4x + 1} \right)dx} = 7 \\ \end{gathered}$