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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.

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

Let I = $integral subscript 0 superscript 8 straight x to the power of 5 over 3 end exponent space dx space minus space open square brackets fraction numerator straight x to the power of begin display style 8 over 3 end style end exponent over denominator begin display style 8 over 3 end style end fraction close square brackets subscript 0 superscript 8 space equals space 3 over 8 open square brackets straight x to the power of 8 over 3 end exponent close square brackets subscript 0 superscript 8$
$equals space 3 over 8 open square brackets left parenthesis 8 right parenthesis to the power of 8 over 3 end exponent minus 0 close square brackets space equals space 3 over 8 left parenthesis 2 to the power of 8 minus 0 right parenthesis space equals space 3 over 8 cross times 256 space equals space 96$

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