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Integrals

Short Answer Type

41.

Evaluate the following definite integral
$integral subscript 0 superscript straight pi over 4 end superscript space sin space 2 straight x space dx$

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42.

Evaluate the following definite integral
$integral subscript straight pi over 6 end subscript superscript straight pi over 4 end superscript space cosec space straight x space dx$

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43.

Evaluate the following definite integral
$integral subscript 4 superscript 5 straight e to the power of straight x dx$

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44.

Evaluate the following definite integral
$integral subscript 0 superscript straight pi over 4 end superscript space left parenthesis 2 sec squared straight x plus straight x cubed plus 2 right parenthesis dx$

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45.
Evaluate the following definite integral
$integral subscript 0 superscript straight pi over 4 end superscript left parenthesis 2 sec squared straight x plus straight x cubed plus 2 right parenthesis space dx$

Let I = $integral subscript 0 superscript straight pi over 4 end superscript left parenthesis 2 sec squared straight x plus straight x cubed plus 2 right parenthesis space dx space equals space open square brackets 2 space tan space straight x space plus straight x to the power of 4 over 4 plus 2 straight x close square brackets subscript 0 superscript straight pi over 4 end superscript$
$equals space open square brackets 2 space tan space straight pi over 4 plus open parentheses begin display style straight pi over 4 end style close parentheses to the power of 4 over 4 plus 2 open parentheses straight pi over 4 close parentheses close square brackets space minus left square bracket 2 space tan space 0 space plus space 0 space plus space 0 right square bracket equals space 2 cross times 1 plus straight pi to the power of 4 over 1024 plus straight pi over 2 minus left parenthesis 0 plus 0 plus 0 right parenthesis space equals space 2 plus straight pi to the power of 4 over 1024 plus straight pi over 2$

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46.

Evaluate the following definite integral
$integral subscript 0 superscript straight pi open parentheses sin squared straight x over 2 minus cos squared straight x over 2 close parentheses space dx$

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47.

Evaluate the following definite integral
$integral subscript 1 superscript 2 left parenthesis 4 straight x cubed minus 5 straight x squared plus 6 straight x plus 9 right parenthesis space dx.$

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48.

Evaluate $integral subscript 0 superscript 1 fraction numerator dx over denominator square root of 1 plus straight x end root minus square root of straight x end fraction.$