Question

Evaluate: $integral subscript negative 5 end subscript superscript 0 straight f left parenthesis straight x right parenthesis space dx comma space space space where space straight f left parenthesis straight x right parenthesis space equals space open vertical bar straight x close vertical bar plus open vertical bar straight x plus 3 close vertical bar plus open vertical bar straight x plus 6 close vertical bar.$

Solution

Here f (x) = | x | + | x + 3 | + | x + 6 |
Let I =

integral subscript negative 5 end subscript superscript 0 straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript negative 5 end subscript superscript negative 3 end superscript straight f left parenthesis straight x right parenthesis space dx space plus space integral subscript negative 3 end subscript superscript 0 straight f left parenthesis straight x right parenthesis space dx

equals space integral subscript negative 5 end subscript superscript negative 3 end superscript open square brackets open vertical bar straight x close vertical bar plus open vertical bar straight x plus 3 close vertical bar plus open vertical bar straight x plus 6 close vertical bar close square brackets space dx space space plus space integral subscript negative 3 end subscript superscript 0 space open square brackets open vertical bar straight x close vertical bar plus open vertical bar straight x plus 3 close vertical bar plus open vertical bar straight x plus 6 close vertical bar close square brackets dx equals integral subscript negative 5 end subscript superscript negative 3 end superscript open square brackets negative straight x minus left parenthesis straight x plus 3 right parenthesis plus left parenthesis straight x plus 6 right parenthesis close square brackets dx plus integral subscript negative 3 end subscript superscript 0 open square brackets negative straight x plus left parenthesis straight x plus 3 right parenthesis plus left parenthesis straight x plus 6 right parenthesis close square brackets dx equals integral subscript negative 5 end subscript superscript negative 3 end superscript left parenthesis negative straight x minus straight x minus 3 plus straight x plus 6 right parenthesis dx plus integral subscript negative 3 end subscript superscript 0 left parenthesis negative straight x plus straight x plus 3 plus straight x plus 6 right parenthesis space dx equals space integral subscript negative 5 end subscript superscript negative 3 end superscript left parenthesis negative straight x plus 3 right parenthesis dx plus integral subscript negative 3 end subscript superscript 0 left parenthesis straight x plus 9 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 negative 3 end superscript space plus space open square brackets straight x squared over 2 plus 9 straight x close square brackets subscript negative 3 end subscript superscript 0

equals space open parentheses negative 9 over 2 minus 9 close parentheses space minus space open parentheses negative 25 over 2 minus 15 close parentheses plus left parenthesis 0 plus 0 right parenthesis minus open parentheses 9 over 2 minus 27 close parentheses minus 27 over 2 plus 55 over 2 plus 45 over 2 space equals space 73 over 2

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