A) \[2\]
B) \[\frac{5}{2}\]
C) \[\frac{1}{2}\]
D) \[\frac{3}{2}\]
Correct Answer: B
Solution :
\[\int_{1}^{4}{\left| x-3 \right|}\,dx\] \[=\int_{1}^{3}{(3-x)}\,dx+\int_{1}^{4}{(x-3)}\,dx\] \[=\left[ 3x-\frac{{{x}^{2}}}{2} \right]_{1}^{3}+\left[ \frac{{{x}^{2}}}{2}-3x \right]_{3}^{4}\] \[=\left( 9-\frac{9}{2} \right)-\left( 3-\frac{1}{2} \right)+\left( \frac{16}{2}-12 \right)-\left( \frac{9}{2}-9 \right)\] \[=\frac{9}{2}-\frac{5}{2}-4+\frac{9}{2}=9-4-\frac{5}{2}\] \[=5-\frac{5}{2}=\frac{5}{2}\]
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