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41.
Evaluate the following definite integral
$integral subscript 0 superscript straight pi over 4 end superscript space sin space 2 straight x space dx$

Let I = $space integral subscript 0 superscript straight pi over 4 end superscript space sin space 2 straight x space dx space equals space open square brackets negative fraction numerator cos space 2 straight x over denominator 2 end fraction close square brackets subscript 0 superscript straight pi over 4 end superscript space equals space minus 1 half open square brackets cos space 2 straight x close square brackets subscript 0 superscript straight pi over 4 end superscript$
$equals negative 1 half left square bracket cos space straight pi over 2 minus cos space 0 right square bracket space equals space minus 1 half left parenthesis 0 minus 1 right parenthesis space equals space 1 half$

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

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