Evaluate:
              \int_{1}^{5}\left ( \left | x-1 \right |+\left | x-2 \right | +\left | x-4 \right |\right )dx

 

 

 

 
 
 
 
 

Answers (1)

\Rightarrow \int_{1}^{5}\left ( \left | x-1 \right |+\left | x-2 \right |+\left | x-4 \right | \right )dx
\Rightarrow \int_{1}^{2}\left ( x-1 \right )-\left ( x-2 \right )-\left ( x-4 \right )dx+\int_{2}^{4}\left ( x-1 \right )+\left ( x-2 \right )-\left ( x-4 \right )dx
                                                                       +\int_{4}^{5}\left ( x-1 \right )+\left ( x-2 \right )+\left ( x-4 \right )dx
\Rightarrow \int_{1}^{2}\left ( 5-x \right )dx+\int_{2}^{4}\left ( x+1 \right )dx+\int_{4}^{5}\left ( 3x-7 \right )dx
\Rightarrow \left [ 5x-\frac{x^{2}}{2} \right ]^{2}_{1}+\left [ \frac{x^{2}}{2}+x \right ]^{4}_{2}+\left [ \frac{3x^{2}}{2}-7x \right ]^{5}_{4}
\Rightarrow \frac{7}{2}+8+\frac{13}{2}= 18        

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