By using the properties of definite integrals, evaluate the integ… - Vidyadip

Question

By using the properties of definite integrals, evaluate the integral $\int\limits_2^8 {\left| {x - 5} \right|dx} $

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Answer

Let $I = \int\limits_2^8 {\left| {x - 5} \right|dx} $…(i)
Putting x - 5 = 0
$ \Rightarrow x = 5 \in \left( {2,8} \right)$
$\therefore$ From eq. (i),
$I = \int\limits_2^5 {\left| {x - 5} \right|dx + \int\limits_5^8 {\left| {x + 5} \right|dx} } $
$= \int\limits_2^5 { - \left( {x - 5} \right)dx + \int\limits_5^8 {\left( {x - 5} \right)dx} } $
$= - \left( {\frac{{{x^2}}}{2} - 5x} \right)_2^5 + \left( {\frac{{{x^2}}}{2} - 5x} \right)_{^5}^8$
$= - \left[ {\left( {\frac{{25}}{2} - 25} \right) - \left( {2 - 10} \right)} \right] + \left[ {\left( {32 - 40} \right) - \left( {\frac{{25}}{2} - 25} \right)} \right]$
$= - \left( { - \frac{{25}}{2} + 8} \right) + \left( { - 8 + \frac{{25}}{2}} \right)$
$= \frac{{25}}{2} - 8 - 8 + \frac{{25}}{2}$
= 25 - 16
= 9

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