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Integrals

Short Answer Type

31.

Evaluate $integral subscript straight a superscript straight b cosx space dx$ as the limit of a sum.

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Long Answer Type

32.
Evaluate $integral subscript straight a superscript straight b space sin squared straight x space dx$ as the limit of a sum.

Comparing $integral subscript straight a superscript straight b space sin squared straight x space dx space space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma space we space get comma$
$straight f left parenthesis straight x right parenthesis space equals space sin squared straight x equals fraction numerator 1 minus cos space 2 straight x over denominator 2 end fraction space equals space 1 half left parenthesis 1 minus cos space 2 straight x right parenthesis$
Now, $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus.... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
$therefore$ $integral subscript straight a superscript straight b sin squared straight x space dx space equals space Lt with straight h rightwards arrow 0 below straight h over 2 left square bracket left parenthesis 1 minus cos space 2 straight a plus left curly bracket 1 minus cos space 2 space left parenthesis straight a plus straight h right parenthesis right curly bracket$
$plus left curly bracket 1 minus cos 2 left parenthesis straight a plus 2 straight h right parenthesis right curly bracket plus.... plus right curly bracket space left curly bracket 1 minus cos space 2 left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right curly bracket right square bracket$
$equals space Lt with straight h rightwards arrow 0 below straight h over 2 left square bracket straight n minus left curly bracket cos 2 straight a plus cos 2 left parenthesis straight a plus straight h right parenthesis plus cos 2 left parenthesis straight a plus 2 straight h right parenthesis plus... plus cos 2 left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right curly bracket right square bracket$
$space space equals space Lt with straight h rightwards arrow 0 below space straight h over 2 open square brackets straight n minus fraction numerator cos space open curly brackets begin display style fraction numerator 2 straight a plus 2 straight a plus 2 left parenthesis straight n minus 1 right parenthesis straight h over denominator 2 end fraction end style close curly brackets sin open parentheses straight n begin display style fraction numerator 2 straight h over denominator 2 end fraction end style close parentheses over denominator sin open parentheses begin display style fraction numerator 2 straight h over denominator 2 end fraction end style close parentheses end fraction close square brackets$

$equals space Lt with straight h rightwards arrow 0 below straight h over 2 open square brackets straight n minus fraction numerator cos left parenthesis 2 straight a plus left parenthesis straight n minus 1 right parenthesis space straight h right curly bracket space sin space straight n space straight h over denominator sin space straight h end fraction close square brackets space equals space Lt with straight h rightwards arrow 0 below 1 half open square brackets straight n space straight h minus fraction numerator cos left curly bracket 2 straight a plus left parenthesis nh minus straight h right parenthesis right curly bracket space sin space nh over denominator begin display style fraction numerator sin space straight h over denominator straight h end fraction end style end fraction close square brackets equals space Lt with straight h rightwards arrow 0 below 1 half open square brackets straight b minus straight a minus fraction numerator cos left curly bracket 2 straight a plus left parenthesis straight b minus straight a minus straight h right parenthesis right curly bracket space sin left parenthesis straight b minus straight a right parenthesis over denominator begin display style fraction numerator sin space straight h over denominator straight h end fraction end style end fraction close square brackets$

$space space equals space 1 half open square brackets straight b minus straight a minus fraction numerator cos left parenthesis 2 straight a plus straight b minus straight a minus 0 right parenthesis space sin left parenthesis straight b minus straight a right parenthesis over denominator 1 end fraction close square brackets space equals space 1 half left square bracket straight b minus straight a minus cos left parenthesis straight a plus straight b right parenthesis space sin left parenthesis straight b minus straight a right parenthesis right square bracket$

$equals space 1 half open square brackets left parenthesis straight b minus straight a right parenthesis space minus space 1 half 2 space cos left parenthesis straight b plus straight a right parenthesis space sin left parenthesis straight b minus straight a right parenthesis right square bracket close square brackets equals space 1 half open square brackets left parenthesis straight b minus straight a right parenthesis minus 1 half left parenthesis sin space 2 straight b space minus space sin space 2 straight a right parenthesis close square brackets equals space 1 half open square brackets left parenthesis straight b minus straight a right parenthesis plus left parenthesis sin space straight a space cos space straight a space minus space sin space straight b space cos space straight b right parenthesis close square brackets$

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Short Answer Type

33.

Evaluate the following definite integrals as limit of sums.
$integral subscript 2 superscript 3 straight x squared dx$

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34.

Evaluate the following definite integral
$integral subscript negative 1 end subscript superscript 1 left parenthesis straight x plus 1 right parenthesis space dx$

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35.

Evaluate the following definite integral
$integral subscript 2 superscript 3 1 over straight x dx$

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36.

Evaluate the following definite integral
$integral subscript 0 superscript 8 straight x to the power of 5 over 3 end exponent dx$

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37.

Evaluate the following definite integral
$integral subscript 0 superscript 4 straight x to the power of 1 half end exponent dx$

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38.

Evaluate the following definite integral
$integral subscript 0 superscript 4 open parentheses straight x plus straight x to the power of 3 over 2 end exponent close parentheses dx$