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Integrals

Question
CBSEENMA12032347

Evaluate the following integral as the limit of a sum:
integral subscript 1 superscript 3 left parenthesis 2 straight x plus 3 right parenthesis dx

Solution

Comparing integral subscript 1 superscript 3 left parenthesis 2 straight x plus 3 right parenthesis dx space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma we get
         f(x) = 2x +3 ,   a = 1,  b = 3
therefore    straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 1 right parenthesis space equals space 2 plus 3 space equals space 5 comma space space 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 space plus space 3 space equals space 5 space plus space 2 straight h
         f(a+2h) = f(1+2h) = 2(1+2h)+3 = 5+4h
          ...     ...      ...        ...         ...          ...
        straight f left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis space equals space straight f left parenthesis 1 plus stack straight n minus 1 with bar on top straight h right parenthesis space equals space 2 open curly brackets 1 plus left parenthesis straight n minus 1 right parenthesis space straight h close curly brackets space plus 3 space equals space 5 plus 2 left parenthesis straight n minus 1 right parenthesis space straight h
Now, integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis dx space equals space Lt with straight h rightwards arrow 0 below 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 space straight h right parenthesis close square brackets
rightwards double arrow space space space integral subscript 1 superscript 3 left parenthesis 2 straight x plus 3 right parenthesis dx space equals space Lt with straight h rightwards arrow 0 below straight h left square bracket 5 plus left parenthesis 5 plus 2 straight h right parenthesis plus left parenthesis 5 plus 4 straight h right parenthesis plus.... plus open curly brackets 5 plus 2 left parenthesis straight n minus 1 right parenthesis space straight h close curly brackets
                                  equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket 5 straight n plus 2 straight h open curly brackets 1 plus 2 plus 3 plus.... plus left parenthesis straight n minus 1 right parenthesis close curly brackets right square bracket
equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets 5 straight n plus 2 straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space straight n over denominator 2 end fraction close square brackets space equals space Lt with straight h rightwards arrow 0 below open square brackets 5 space straight n space straight h space plus left parenthesis straight n space straight h right parenthesis space left parenthesis nh space minus straight h right parenthesis close square brackets
equals space Lt with straight h rightwards arrow 0 below space left square bracket space 5 space left parenthesis 2 right parenthesis space plus space left parenthesis 2 right parenthesis space left parenthesis 2 minus straight h right parenthesis right square bracket space space space space space space space space space space space space space space space space open square brackets because space straight n space straight h space space equals space straight b space minus straight a space equals space 3 space minus space 1 space equals space 2 close square brackets
space equals space 5 space left parenthesis 2 right parenthesis space plus space 2 left parenthesis 2 minus 0 right parenthesis space equals space 10 plus 4 space equals space 14

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