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

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

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39.
Evaluate the following definite integral
$integral subscript 0 superscript straight pi space tanx space dx$

Let I = $integral subscript 0 superscript straight pi space tanx space dx space equals space left square bracket log open vertical bar cosx close vertical bar right square bracket subscript 0 superscript straight pi space equals space minus left square bracket log open vertical bar cos space straight pi close vertical bar space minus space left square bracket log open vertical bar cos space straight pi close vertical bar space minus log open vertical bar cos space 0 close vertical bar right square bracket$
$equals space left square bracket negative log space 1 space space minus space log space 1 right square bracket space equals space 0$