Evaluate as limit of sums: $integral subscript 1 superscript 3 left parenthesis 2 straight x squared plus 3 straight x right parenthesis space dx. space$

Solution

Comparing $integral subscript 1 superscript 3 left parenthesis 2 straight x squared plus 3 straight x right parenthesis space dx space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma space we space get$
   $straight f left parenthesis straight x right parenthesis space equals space 2 straight x squared plus 3 straight x comma space space straight a space equals space 1 comma space space space straight b space equals space 3 straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 1 right parenthesis space equals space 2 space plus 3 straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis 1 plus straight h right parenthesis space equals space 2 left parenthesis 1 plus straight h right parenthesis squared plus 3 left parenthesis 1 plus straight h right parenthesis straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight f left parenthesis 1 plus 2 straight h right parenthesis space equals 2 left parenthesis 1 plus 2 straight h right parenthesis squared plus 3 left parenthesis 1 plus 2 straight h right parenthesis$
.................................................................
   $straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space straight f left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space 2 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared plus 3 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis$
Now,    $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus.... plus straight f left parenthesis straight a plus top enclose straight n minus 1 end enclose right parenthesis space straight h right parenthesis close square brackets$
   $equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket open curly brackets 2 plus 3 close curly brackets space plus space left curly bracket 2 space left parenthesis 1 plus straight h right parenthesis squared space plus space 3 space left parenthesis 1 plus straight h right parenthesis right curly bracket space plus space left curly bracket 2 left parenthesis 1 plus 2 straight h right parenthesis squared plus 3 left parenthesis 1 plus 2 straight h right parenthesis space right curly bracket$
   $plus.... plus left curly bracket 2 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared space plus space 3 space left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis right curly bracket right square bracket$
   $equals space Lt with straight h rightwards arrow 0 below left square bracket 2 space straight n space plus space 3 space straight n space plus space 4 space straight h space left square bracket 1 plus 2 plus........ plus stack straight n minus 1 with bar on top right curly bracket space plus space 3 space straight h space left curly bracket 1 plus 2 plus.... plus stack straight n minus 1 with bar on top right curly bracket$
   $plus 2 space straight h squared space left curly bracket 1 squared plus 2 squared plus......... left parenthesis stack straight n minus 1 with bar on top right parenthesis squared right curly bracket right square bracket$
   $space equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets 5 space straight n plus space 4 space straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space straight n over denominator 2 end fraction plus 3 space straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis over denominator 2 end fraction plus 2 space straight h squared space fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis space left parenthesis 2 straight n minus 1 right parenthesis over denominator 6 end fraction close square brackets equals space Lt with straight h rightwards arrow 0 below space open square brackets 5 space straight n space straight h space plus space 2 space left parenthesis space straight n space straight h space minus straight h right parenthesis space left parenthesis space straight n space straight h right parenthesis space plus space 3 over 2 left parenthesis straight n space straight h space minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis space plus space 1 third left parenthesis space straight n space straight h space minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis thin space left parenthesis 2 space straight n space straight h space minus straight h right parenthesis close square brackets$
   $equals space Lt with straight h rightwards arrow 0 below space open square brackets 5 cross times 2 plus 2 left parenthesis 2 minus straight h right parenthesis space left parenthesis 2 right parenthesis space plus space 3 over 2 left parenthesis 2 minus straight h right parenthesis space left parenthesis 2 right parenthesis space plus space 1 third left parenthesis 2 minus straight h right parenthesis space left parenthesis 2 right parenthesis space left parenthesis 4 minus straight h right parenthesis close square brackets$
   $left square bracket because space straight n space straight h space equals space space straight b space minus space straight a space equals space 3 space minus space 1 space equals space 2 space right square bracket$
$equals 10 plus 4 space left parenthesis 2 minus 0 right parenthesis space plus space 3 over 2 left parenthesis 2 minus 0 right parenthesis space left parenthesis 2 right parenthesis space plus 1 third left parenthesis 2 minus 0 right parenthesis space left parenthesis 2 right parenthesis space left parenthesis 4 minus 0 right parenthesis equals space 10 plus 8 plus 6 plus 16 over 3 space equals space 88 over 3$

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Integrals

Evaluate as limit of sums: $integral subscript 1 superscript 3 left parenthesis 2 straight x squared plus 3 straight x right parenthesis space dx. space$

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