By using the properties of definite integrals, evaluate the integral $\int_{2}^{8} | x - 5 | d x$

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Solution

Let $I = \int_{2}^{8} | x - 5 | d x$
It can be seen that $(x - 5) \leq 0$ on $[2, 5]$ and $(x - 5) \geq 0$ on $[5, 8]$ .
$∴ I = \int_{2}^{5} - (x - 5) d x + \int_{1}^{8} (x - 5) d x$
$= - {[\frac{x^{2}}{2} - 5 x]}_{2}^{5} + {[\frac{x^{2}}{2} - 5 x]}_{2}^{8}$
$= - [\frac{25}{2} - 25 - 2 + 10] + [32 - 40 - \frac{25}{2} + 25]$
$= 9$

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