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Integrals

Question

Wired Faculty App

Evaluate $integral subscript negative 1 end subscript superscript 2 left parenthesis straight e to the power of 3 straight x end exponent plus 7 straight x minus 5 right parenthesis dx$

Solution

integral subscript negative 1 end subscript superscript 2 left parenthesis straight e to the power of 3 straight x end exponent plus 7 straight x minus 5 right parenthesis dx Here space straight f left parenthesis straight x right parenthesis space equals space straight e to the power of 3 straight x end exponent plus 7 straight x minus 5 straight a space equals negative 1 comma space space straight b equals space 2 comma space space straight h space equals space fraction numerator straight b minus straight a over denominator straight n end fraction equals 3 over straight n By space definition space integral subscript negative 1 end subscript superscript 2 left parenthesis straight e to the power of 3 straight x end exponent plus 7 straight x minus 5 right parenthesis dx space equals limit as straight n rightwards arrow infinity of sum from straight r equals 1 to straight n of straight h. straight f left parenthesis straight a plus rh right parenthesis limit as straight n rightwards arrow infinity of sum from straight r equals 1 to straight n of straight h. straight f left parenthesis negative 1 plus rh right parenthesis equals limit as straight n rightwards arrow infinity of sum from straight r equals 1 to straight n of straight h. open parentheses straight e to the power of 3 left parenthesis negative 1 plus rh right parenthesis end exponent plus 7 left parenthesis negative 1 plus rh right parenthesis minus 5 close parentheses space space space equals space limit as straight n rightwards arrow infinity of open square brackets straight h. straight e to the power of negative 3 end exponent. space straight e to the power of 3 straight h end exponent open parentheses 1 plus straight e to the power of 3 straight h end exponent plus straight e to the power of 6 straight h end exponent plus.. plus straight e to the power of 3 nh end exponent close parentheses plus 7 straight h squared left parenthesis 1 plus 2 plus 3 plus.. plus straight n right parenthesis minus 12 nh close square brackets space space space equals space limit as straight n rightwards arrow infinity of open square brackets he to the power of 3 straight h end exponent over straight e cubed cross times fraction numerator straight e to the power of 3 nh end exponent minus 1 over denominator straight e to the power of 3 straight h end exponent minus 1 end fraction plus 7 straight h squared fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction minus 12 nh close square brackets space space space equals limit as straight n rightwards arrow infinity of open square brackets open parentheses fraction numerator 3 straight e to the power of 3 cross times begin display style 3 over straight n end style end exponent over denominator ne cubed end fraction cross times open parentheses straight e to the power of 3 straight n cross times 3 over straight n minus 1 end exponent close parentheses cross times open parentheses fraction numerator 3 straight h over denominator straight e to the power of 3 straight h end exponent minus 1 end fraction close parentheses cross times fraction numerator straight n over denominator 3 cross times 3 end fraction close parentheses plus 63 over straight n squared cross times fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction minus 12 cross times 3 close square brackets

Now applying the limit we get,

equals fraction numerator straight e to the power of 9 minus 1 over denominator 3 straight e cubed end fraction plus 63 over 2 minus 36 equals fraction numerator straight e to the power of 9 minus 1 over denominator 3 straight e cubed end fraction minus 9 over 2

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