Evaluate as limit of sums $integral subscript 1 superscript 4 left parenthesis 5 straight x squared plus 3 straight x right parenthesis space dx.$

Solution

Comparing $integral subscript 1 superscript 4 space left parenthesis 5 straight x squared plus 3 straight x right parenthesis space dx space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma space space we space get$
$space straight f left parenthesis straight x right parenthesis space equals space 5 straight x squared plus 3 straight x comma space space space straight a space equals space 1 comma space space space straight b space equals space 4 straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 1 right parenthesis space equals space 5 plus 3 straight f left parenthesis straight a plus straight b right parenthesis space equals space straight f left parenthesis 1 plus straight h right parenthesis space equals space 5 left parenthesis 1 plus straight h right parenthesis squared plus 3 left parenthesis 1 plus straight h right parenthesis straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight f left parenthesis 1 plus 2 straight h right parenthesis space equals space 5 left parenthesis 1 plus 2 straight h right parenthesis squared plus 3 left parenthesis 1 plus 2 straight h right parenthesis$
............................................................................
$straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space straight f left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space 5 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared space plus space 3 left parenthesis 1 plus stack straight n minus 1 with bar on top straight h right parenthesis$
Now, $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus.... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top right parenthesis straight h right square bracket$
$equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket left curly bracket 5 plus 3 right curly bracket space plus space left curly bracket 5 space left parenthesis 1 plus straight h right parenthesis squared plus 3 left parenthesis 1 plus straight h right parenthesis right curly bracket space plus space left curly bracket 5 left parenthesis 1 plus 2 straight h right parenthesis squared plus 3 left parenthesis 1 plus 2 straight h right parenthesis right curly bracket$
   $plus........... plus left curly bracket 5 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared plus 3 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis right curly bracket right square bracket$
   $equals Lt with straight h rightwards arrow 0 below space straight h space left square bracket 5 straight n space plus 3 straight n plus 10 space straight h space left curly bracket 1 plus 2 plus... plus stack straight n minus 1 with bar on top right curly bracket space plus space 3 space straight h space left curly bracket 1 plus 2 plus.... plus stack straight n minus 1 with bar on top right curly bracket$
$plus space 5 space straight h squared space left curly bracket 1 squared plus 2 squared plus..... open parentheses stack straight n minus 1 with bar on top close parentheses squared right curly bracket right square bracket$
   $space equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets space 8 space straight n space plus space 10 space straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis over denominator 2 end fraction plus 3 space straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis over denominator 2 end fraction plus fraction numerator 5 space straight h squared space left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis space left parenthesis 2 straight n minus 1 right parenthesis over denominator 6 end fraction close square brackets equals space space Lt with straight h rightwards arrow 0 below space open square brackets 8 space straight n space straight h space plus space 5 space left parenthesis straight n space straight h space minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis space plus space 3 over 2 left parenthesis space straight n space straight h space minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis space plus space 5 over 6 left parenthesis space straight n space straight h space minus space straight h right parenthesis space left parenthesis space straight n space straight h right parenthesis space left parenthesis 2 space straight n space straight h space minus space straight h right parenthesis close square brackets$
   $equals space Lt with straight h rightwards arrow 0 below space open square brackets 8 plus 3 plus 5 left parenthesis 3 minus straight h right parenthesis space left parenthesis 3 right parenthesis space plus space 3 over 2 left parenthesis 3 minus straight h right parenthesis thin space left parenthesis 3 right parenthesis space plus space 5 over 6 left parenthesis 3 minus straight h right parenthesis space left parenthesis 3 right parenthesis space left parenthesis 6 minus straight h right parenthesis close square brackets$
   $open square brackets because space straight n space straight h space equals space straight b space minus straight a space equals space 4 minus 1 space equals 3 close square brackets$
   $equals space 24 plus 15 left parenthesis 3 minus 0 right parenthesis space plus space 9 over 2 left parenthesis 3 minus 0 right parenthesis space plus space 5 over 2 left parenthesis 3 minus 0 right parenthesis space left parenthesis 6 minus 0 right parenthesis equals space 24 plus 45 plus 27 over 2 plus 45 space equals space 114 plus 27 over 2 space equals space 255 over 2$

For Daily Free Study Material Join wiredfaculty

Integrals

Evaluate as limit of sums $integral subscript 1 superscript 4 left parenthesis 5 straight x squared plus 3 straight x right parenthesis space dx.$

Some More Questions From Integrals Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction