Question

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 2 superscript 8 open vertical bar straight x minus 5 close vertical bar space dx$

Solution

Let I = $integral subscript 2 superscript 8 open vertical bar straight x minus 5 close vertical bar space dx$
For 2 ≤ r ≤ 5, x – 5 ≤ 0 ⇒ | x – 5 | = – (x – 5) and for 5 ≤ x ≤ 8, x – 5 ≥ 0 ⇒ | x – 5 | = x – 5
$therefore space space space space straight I space equals space integral subscript 2 superscript 5 open vertical bar straight x minus 5 close vertical bar dx space plus space integral subscript 5 superscript 8 open vertical bar straight x minus 5 close vertical bar space dx$
$equals negative integral subscript 2 superscript 5 left parenthesis straight x minus 5 right parenthesis dx plus integral subscript 5 superscript 8 left parenthesis straight x minus 5 right parenthesis dx space equals space minus open square brackets straight x squared over 2 minus 5 straight x close square brackets subscript 2 superscript 5 plus open square brackets straight x squared over 2 minus 5 straight x close square brackets subscript 5 superscript 8$
$equals negative open square brackets open parentheses 25 over 2 minus 25 close parentheses space minus space open parentheses 4 over 2 minus 10 close parentheses close square brackets space plus space open square brackets open parentheses 64 over 2 minus 40 close parentheses minus open parentheses 25 over 2 minus 25 close parentheses close square brackets equals negative open square brackets negative 25 over 2 plus 8 close square brackets space plus open square brackets negative 8 plus 25 over 2 close square brackets space equals space 9 over 2 plus 9 over 2 space equals 9$

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Integrals

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 2 superscript 8 open vertical bar straight x minus 5 close vertical bar space dx$

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