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Q. The value of integral $ \int_{0}^{4}{|x-1|}dx $ is

Rajasthan PETRajasthan PET 2009

Solution:

$ \int_{0}^{4}{|x-1|dx=\int_{0}^{1}{(1-x)dx+\int_{1}^{4}{(x-1)dx}}} $
$ =\left[ x-\frac{{{x}^{2}}}{2} \right]_{0}^{1}+\left[ \frac{{{x}^{2}}}{2}-x \right]_{1}^{4} $
$ =1-\frac{1}{2}+\left[ 8-4-\left( \frac{1}{2}-1 \right) \right] $
$ =\frac{1}{2}+4+\frac{1}{2}=5 $