Question

Evaluate $integral subscript 1 superscript 4 straight f left parenthesis straight x right parenthesis space dx comma space space space where space space space straight f left parenthesis straight x right parenthesis space equals space open curly brackets table attributes columnalign left end attributes row cell 4 straight x plus 3 space if space 1 space less or equal than space straight x space less or equal than space 2 end cell row cell 3 straight x plus 5 space if space 2 space less or equal than space straight x space less or equal than space 4 end cell end table close.$

Solution

Here $straight f left parenthesis straight x right parenthesis space equals space open curly brackets table attributes columnalign left end attributes row cell 4 straight x plus 3 space if space 1 space less or equal than space straight x space less or equal than 2 end cell row cell 3 straight x plus 5 space if space 2 space less or equal than space straight x space less or equal than 4 end cell end table close$
Let I = $integral subscript 1 superscript 4 straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 1 superscript 2 straight f left parenthesis straight x right parenthesis space dx space plus space integral subscript 2 superscript 4 straight f left parenthesis straight x right parenthesis space dx$
$equals space integral subscript 1 superscript 2 left parenthesis 4 straight x plus 3 right parenthesis space dx plus integral subscript 2 superscript 4 space left parenthesis 3 straight x plus 5 right parenthesis space dx space equals space open square brackets 4 straight x squared over 2 plus 3 space straight x close square brackets subscript 1 superscript 2 plus open square brackets fraction numerator 3 straight x squared over denominator 2 end fraction plus 5 straight x close square brackets subscript 2 superscript 4 equals space open square brackets left parenthesis 8 plus 6 right parenthesis space minus space left parenthesis 2 plus 3 right parenthesis close square brackets space plus space open square brackets left parenthesis 24 plus 20 right parenthesis space minus space left parenthesis 6 plus 10 right parenthesis close square brackets equals space left parenthesis 14 minus 5 right parenthesis space plus space left parenthesis 44 minus 16 right parenthesis space equals space 9 space plus space 28 space equals space 37.$

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