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Integrals

Question
CBSEENMA12032651

Evaluate  integral subscript 1 superscript 4 straight f left parenthesis straight x right parenthesis space dx space space where space straight f left parenthesis straight x right parenthesis space equals space open vertical bar straight x minus 1 close vertical bar space plus space open vertical bar straight x minus 2 close vertical bar space plus space open vertical bar straight x minus 3 close vertical bar

Solution

Here,        straight f left parenthesis straight x right parenthesis space equals space open vertical bar straight x minus 1 close vertical bar space plus space open vertical bar straight x minus 2 close vertical bar plus space open vertical bar straight x minus 3 close vertical bar
Let   I = integral subscript 1 superscript 4 straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 1 superscript 2 straight f left parenthesis straight x right parenthesis space dx plus integral subscript 2 superscript 3 straight f left parenthesis straight x right parenthesis space dx plus integral subscript 3 superscript 4 straight f left parenthesis straight x right parenthesis space dx
           equals space integral subscript 1 superscript 2 open square brackets left parenthesis straight x minus 1 right parenthesis space minus space left parenthesis straight x minus 2 right parenthesis space minus space left parenthesis straight x minus 3 right parenthesis close square brackets dx space plus space integral subscript 2 superscript 3 open square brackets left parenthesis straight x minus 1 right parenthesis space plus left parenthesis straight x minus 2 right parenthesis space minus space left parenthesis straight x minus 3 right parenthesis close square brackets dx
                                                                          plus integral subscript 3 superscript 4 open square brackets left parenthesis straight x minus 1 right parenthesis plus left parenthesis straight x minus 2 right parenthesis plus left parenthesis straight x minus 3 right parenthesis close square brackets dx
           equals space integral subscript 1 superscript 2 left parenthesis negative straight x plus 4 right parenthesis space dx plus space integral subscript 2 superscript 3 straight x space dx space plus integral subscript 3 superscript 4 left parenthesis 3 straight x minus 6 right parenthesis space dx
equals space open square brackets negative straight x squared over 2 plus 4 straight x close square brackets subscript 1 superscript 2 space plus space open square brackets straight x squared over 2 close square brackets subscript 2 superscript 3 plus open square brackets fraction numerator 3 straight x squared over denominator 2 end fraction minus 6 straight x close square brackets subscript 3 superscript 4
equals left parenthesis negative 2 plus 8 right parenthesis space space minus space open parentheses negative 1 half plus 4 close parentheses plus open parentheses 9 over 2 minus 2 close parentheses space plus space left parenthesis 24 minus 24 right parenthesis space minus space open parentheses 27 over 2 minus 18 close parentheses
equals space 6 minus 7 over 2 plus 5 over 2 plus 0 plus 9 over 2 space equals space 19 over 2

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