Evaluate $integral subscript 1 superscript 2 space left parenthesis 3 straight x squared plus 5 straight x right parenthesis space dx$ as limit of sums.

Solution

Comparing $integral subscript 1 superscript 2 left parenthesis 3 space straight x squared plus 5 straight x right parenthesis space dx space with space integral subscript straight a superscript straight b space straight f left parenthesis straight x right parenthesis space dx comma space$ we get
   $straight f left parenthesis straight x right parenthesis space equals space 3 straight x squared plus 5 straight x comma space space straight a space equals space 1 comma space space straight b space equals space 2 straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 1 right parenthesis space equals space 3 plus 5 straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis 1 plus straight h right parenthesis space equals space 3 left parenthesis 1 plus straight h right parenthesis squared plus 5 left parenthesis 1 plus straight h right parenthesis straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight f left parenthesis 1 plus 2 straight h right parenthesis space equals space 3 left parenthesis 1 plus 2 straight h right parenthesis squared plus 5 left parenthesis 1 plus 2 straight h right parenthesis$
....................................................................
$straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space straight f left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space 3 open curly brackets 1 plus left parenthesis straight n minus 1 right parenthesis space straight h close curly brackets squared plus 5 open curly brackets 1 plus left parenthesis straight n minus 1 right parenthesis space straight h close curly brackets$
Now $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus..... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis$
$rightwards double arrow$    $integral subscript 1 superscript 2 left parenthesis 3 straight x squared plus 5 straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket open curly brackets 3 plus 5 close curly brackets plus open curly brackets 3 space left parenthesis 1 plus straight h right parenthesis squared space plus space 5 space left parenthesis 1 plus straight h right parenthesis close curly brackets space space space space space space space space space space$
$plus open curly brackets 3 space left parenthesis 1 plus 2 space straight h right parenthesis squared space plus space 5 left parenthesis 1 plus 2 straight h right parenthesis close curly brackets space plus space....... plus open curly brackets 3 space left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared plus 5 left parenthesis 1 plus stack straight n minus 1 with bar on top straight h right parenthesis close curly brackets right square bracket$
   $equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket left parenthesis 3 space straight n plus space 5 space straight n space plus space 6 space straight h space left parenthesis 1 plus 2 plus..... plus stack straight n minus 1 with bar on top right parenthesis plus 5 space straight h space left parenthesis 1 plus 2 plus.... plus stack straight n minus 1 with bar on top right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space$
$plus space 3 space straight h squared space open curly brackets 1 squared plus 2 squared plus.... plus left parenthesis straight n minus 1 right parenthesis squared close curly brackets right square bracket$
$equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets 8 space straight n space plus 6 space straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis over denominator straight n end fraction plus fraction numerator space 5 space straight h space left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis over denominator 2 end fraction plus space 3 space straight h squared space fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis space left parenthesis 2 straight n minus 1 right parenthesis over denominator 6 end fraction close square brackets$
$equals space Lt with straight h rightwards arrow 0 below open square brackets 8 space straight n space straight h space plus space 3 space left parenthesis straight n space straight h right parenthesis space left parenthesis straight n space straight h space minus straight h right parenthesis space space plus 5 over 2 left parenthesis space straight n space straight h right parenthesis space left parenthesis nh space minus space straight h right parenthesis space plus space 1 half left parenthesis space straight n space straight h space minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis space left parenthesis 2 space straight n space straight h space minus space straight h right parenthesis close square brackets$
$equals space Lt with straight h rightwards arrow 0 below open square brackets 8 left parenthesis 1 right parenthesis space plus space 3 space left parenthesis 1 right parenthesis space plus space left parenthesis 1 minus straight h right parenthesis space plus space 5 over 2 left parenthesis 1 right parenthesis space left parenthesis 1 minus straight h right parenthesis space plus 1 half left parenthesis 1 minus straight h right parenthesis space left parenthesis 1 right parenthesis space left parenthesis 2 minus straight h right parenthesis close square brackets$
$open square brackets because space space straight n space straight h space equals space 2 minus 1 space equals space 1 close square brackets$
= $8 plus 3 space left parenthesis 1 minus 0 right parenthesis space plus space 5 over 2 left parenthesis 1 minus 0 right parenthesis space plus space 1 half left parenthesis 1 minus 0 right parenthesis space left parenthesis 2 minus 0 right parenthesis space equals space 8 plus 3 plus 5 over 2 plus 1 space equals space 29 over 2$

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Integrals

Evaluate $integral subscript 1 superscript 2 space left parenthesis 3 straight x squared plus 5 straight x right parenthesis space dx$ as limit of sums.

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