Evaluate $integral subscript 0 superscript 4 left parenthesis straight x plus straight e to the power of 2 straight x end exponent right parenthesis space dx$ as the limit of a sum.

Solution

Comparing $integral subscript 0 superscript 4 left parenthesis straight x plus straight e to the power of 2 straight x end exponent right parenthesis dx space space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis comma space we space get$
$straight f left parenthesis straight x right parenthesis space equals space straight x plus straight e to the power of 2 straight x end exponent comma space space straight a space equals space 0 comma space space straight b space equals space 4$
Now, $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
$therefore$ $integral subscript 0 superscript 4 left parenthesis straight x plus straight e to the power of 2 straight x end exponent right parenthesis space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight f left parenthesis 0 right parenthesis plus straight f left parenthesis straight h right parenthesis plus straight f left parenthesis 2 straight h right parenthesis plus... plus straight f left parenthesis stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
$equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket left parenthesis 0 plus straight e to the power of 0 right parenthesis space plus left parenthesis straight h plus straight e to the power of 2 straight h end exponent right parenthesis space plus space left parenthesis 2 straight h plus straight e to the power of 4 straight h end exponent right parenthesis plus... plus left curly bracket left parenthesis straight n minus 1 right parenthesis space straight h space plus straight e to the power of 2 left parenthesis straight n minus 1 right parenthesis straight h end exponent right curly bracket right square bracket$
$equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket straight h left parenthesis 1 plus 2 plus 3 plus... plus stack straight n minus 1 with bar on top right parenthesis plus left parenthesis 1 plus straight e to the power of 2 straight h end exponent plus straight e to the power of 4 straight h end exponent plus... plus straight e to the power of 2 left parenthesis straight n minus 1 right parenthesis space straight h end exponent right parenthesis right square bracket$
$equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis over denominator 2 end fraction plus fraction numerator 1 left parenthesis straight e to the power of 2 nh end exponent minus 1 right parenthesis over denominator straight e to the power of 2 straight h end exponent minus 1 end fraction close square brackets space space space space space space space space space space space space space open square brackets because straight S subscript straight n space equals space fraction numerator straight a left parenthesis straight r to the power of straight n minus 1 right parenthesis over denominator straight r minus 1 end fraction comma space straight r greater than 1 close square brackets equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets fraction numerator left parenthesis straight n space straight h minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis over denominator 2 end fraction plus fraction numerator straight e to the power of 2 nh end exponent minus 1 over denominator 2 cross times begin display style fraction numerator straight e to the power of 2 straight h end exponent minus 1 over denominator 2 straight h end fraction end style end fraction close square brackets space equals space Lt with straight h rightwards arrow 0 below space open square brackets fraction numerator left parenthesis 4 minus straight h right parenthesis space left parenthesis 4 right parenthesis over denominator 2 end fraction plus fraction numerator straight e to the power of 8 minus 1 over denominator 2 cross times begin display style fraction numerator straight e to the power of 2 straight h end exponent minus 1 over denominator 2 straight h end fraction end style end fraction close square brackets$
$open square brackets because space straight n apostrophe straight h space equals space straight b minus straight a space equals space 4 minus 0 space equals space 4 close square brackets$
= $fraction numerator left parenthesis 4 minus 0 right parenthesis space left parenthesis 4 right parenthesis over denominator 2 end fraction plus fraction numerator straight e to the power of 8 minus 1 over denominator 2 cross times loge end fraction space equals space 8 plus fraction numerator straight e to the power of 8 minus 1 over denominator 2 end fraction space equals space fraction numerator straight e to the power of 8 plus 15 over denominator 2 end fraction$

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Integrals

Evaluate $integral subscript 0 superscript 4 left parenthesis straight x plus straight e to the power of 2 straight x end exponent right parenthesis space dx$ as the limit of a sum.

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