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

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

Let
$straight I space equals space integral subscript 0 superscript 1 fraction numerator 1 over denominator square root of 1 plus straight x end root minus square root of straight x end fraction dx space equals space integral subscript 0 superscript 1 fraction numerator 1 over denominator square root of 1 plus straight x end root minus square root of straight x end fraction cross times fraction numerator square root of 1 plus straight x end root plus square root of straight x over denominator square root of 1 plus straight x end root plus square root of straight x end fraction dx space space space$

$space equals space integral subscript 0 superscript 1 fraction numerator square root of 1 plus straight x end root plus square root of straight x over denominator left parenthesis 1 plus straight x right parenthesis minus straight x end fraction dx space equals space integral subscript 0 superscript 1 fraction numerator square root of 1 plus straight x end root plus square root of straight x over denominator 1 end fraction dx$
$equals integral subscript 0 superscript 1 space open square brackets left parenthesis plus straight x right parenthesis to the power of 1 half end exponent plus straight x to the power of 1 half end exponent close square brackets space dx space space equals space open square brackets fraction numerator left parenthesis 1 plus straight x right parenthesis to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction plus fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 0 superscript 1$
$equals space 2 over 3 open square brackets left parenthesis 1 plus straight x right parenthesis to the power of 3 over 2 end exponent plus straight x to the power of 3 over 2 end exponent close square brackets subscript 0 superscript 1 space equals space 2 over 3 open square brackets open parentheses 2 to the power of 3 over 2 end exponent plus 1 to the power of 3 over 2 end exponent close parentheses minus left parenthesis 1 plus 0 right parenthesis close square brackets equals space 2 over 3 left square bracket 2 square root of 2 plus 1 minus 1 right square bracket space equals space 2 over 3 cross times 2 square root of 2 space equals space fraction numerator 4 square root of 2 over denominator 3 end fraction$