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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.
$integral subscript 2 superscript 3 straight x squared dx$

Let I = $integral subscript 2 superscript 3 straight x squared dx$
Since $integral straight x squared dx space equals space straight x cubed over 3 space equals space straight F left parenthesis straight x right parenthesis comma space say.$
$therefore$ by second fundamental theorem,
I = F(3) - F(2) = $fraction numerator left parenthesis 3 right parenthesis cubed over denominator 3 end fraction minus fraction numerator left parenthesis 2 right parenthesis cubed over denominator 3 end fraction space equals space 9 minus 8 over 3 space equals space 19 over 3$

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