Question

Evaluate: $integral subscript negative 5 end subscript superscript 0 space straight f left parenthesis straight x right parenthesis space dx comma space space where space straight f left parenthesis straight x right parenthesis space equals space open vertical bar straight x close vertical bar plus open vertical bar straight x plus 2 close vertical bar plus open vertical bar straight x plus 5 close vertical bar.$

Solution

Here $straight f left parenthesis straight x right parenthesis space equals space open vertical bar straight x close vertical bar plus open vertical bar straight x plus 2 close vertical bar plus open vertical bar straight x plus 5 close vertical bar$
Let I = $integral subscript negative 5 end subscript superscript 0 space straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript negative 5 end subscript superscript negative 2 end superscript straight f left parenthesis straight x right parenthesis space dx space plus space integral subscript negative 2 end subscript superscript 0 space straight f left parenthesis straight x right parenthesis space dx$
$equals space integral subscript negative 5 end subscript superscript negative 2 end superscript open square brackets open vertical bar straight x close vertical bar plus open vertical bar straight x plus 2 close vertical bar plus open vertical bar straight x plus 5 close vertical bar close square brackets dx plus integral subscript negative 2 end subscript superscript 0 open square brackets open vertical bar straight x close vertical bar plus open vertical bar straight x plus 2 close vertical bar space plus open vertical bar straight x plus 5 close vertical bar close square brackets dx$
$equals space integral subscript negative 5 end subscript superscript negative 2 end superscript left square bracket negative straight x minus left parenthesis straight x plus 2 right parenthesis plus left parenthesis straight x plus 5 right parenthesis right square bracket space dx space plus space integral subscript negative 2 end subscript superscript 0 open square brackets negative straight x plus left parenthesis straight x plus 2 right parenthesis plus left parenthesis straight x plus 5 right parenthesis close square brackets space dx equals space integral subscript negative 5 end subscript superscript negative 2 end superscript left parenthesis negative straight x minus straight x minus 2 plus straight x plus 5 right parenthesis space dx plus integral subscript negative 2 end subscript superscript 0 left parenthesis negative straight x plus straight x plus 2 plus straight x plus 5 right parenthesis space dx equals space integral subscript negative 5 end subscript superscript negative 2 end superscript left parenthesis negative straight x plus 3 right parenthesis space dx plus integral subscript negative 2 end subscript superscript 0 left parenthesis straight x plus 7 right parenthesis space dx space equals space open square brackets negative straight x squared over 2 plus 3 straight x close square brackets subscript negative 5 end subscript superscript 2 plus open square brackets straight x squared over 2 plus 7 straight x close square brackets subscript negative 2 end subscript superscript 0 equals left parenthesis negative 2 minus 6 right parenthesis minus open parentheses negative 25 over 2 minus 15 close parentheses plus left parenthesis 0 plus 0 right parenthesis minus left parenthesis 2 minus 14 right parenthesis space equals space minus 8 plus 55 over 2 plus 12 space equals 63 over 2.$

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