Question

\int _3^6\:|12x-10|dx

Answer 1

Solution

$\int\limits_3^6 {\left| {12x - 10} \right|} \,dx = 2\int\limits_3^6 {\left| {6x - 5} \right|} \,dx$

Then 

$\int {\left| {6x - 5} \right|} \,dx = \frac{1}{6}\int {\left| u \right|} \,du\\ = \frac{1}{6}\frac{{u\left| u \right|}}{2} + C\\ = \frac{{u\left| u \right|}}{{12}} + C\\ = \frac{{\left( {6x - 5} \right)\left| {6x - 5} \right|}}{{12}} + C$

Therefore, 

$\begin{array}{l} \int\limits_3^6 {\left| {12x - 10} \right|} \,dx = \left[ {2 \times \frac{{\left( {6x - 5} \right)\left| {6x - 5} \right|}}{{12}}} \right]_3^6\\ = \frac{1}{6}\left[ {\left( {6x - 5} \right)\left| {6x - 5} \right|} \right]_3^6\\ = \frac{1}{6}\left[ {\left( {6\left( 6 \right) - 5} \right)\left| {6\left( 6 \right) - 5} \right| - \left( {6\left( 3 \right) - 5} \right)\left| {6\left( 3 \right) - 5} \right|} \right]\\ = \frac{1}{6}\left[ {\left( {31} \right)\left| {31} \right| - \left( {18} \right)\left| {18} \right|} \right]\\ = 132 \end{array}$