Integrals — Class 12 Maths Solution

Let$I = \int\limits_2^8 {\left| {x - 5} \right|} dx = \int\limits_2^5 {\left| {x - 5} \right|dx} + \int\limits_2^8 {\left| {x - 5} \right|} dx$

$\left| {x - 5} \right| = \left\{ \begin{array}{l} - \left( {x - 5} \right),\,\,\,\,\,\,{\rm{if}}\,x < 5\\\left( {x - 5} \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{if}}\,x \ge 5\end{array} \right.$

$\therefore$ $I = - \int\limits_2^5 {\left( {x - 5} \right)dx} + \int\limits_5^8 {\left( {x - 5} \right)dx}$

$= - \left[ {\cfrac{{{x^2}}}{2} - 5x} \right]_2^5 + \left[ {\cfrac{{{x^2}}}{2} - 5x} \right]_5^8$

$= - \cfrac{1}{2}\left( {25 - 4} \right) + 5\left( {5 - 2} \right) + \cfrac{1}{2}\left( {64 - 25} \right) - 5\left( {8 - 5} \right)$

$= \cfrac{{ - 21}}{2} + 15 + \cfrac{{39}}{2} - 15 = \cfrac{{18}}{2} = 9$