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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
$integral subscript straight pi over 6 end subscript superscript straight pi over 4 end superscript space cosec space straight x space dx$

Let I = $integral subscript straight pi over 6 end subscript superscript straight pi over 4 end superscript space cos e c space x space d x space equals subscript blank superscript blank$ $open square brackets table row log cell open vertical bar cosecx minus cotx close vertical bar end cell end table close square brackets subscript straight pi over 6 end subscript superscript straight pi over 4 end superscript$
$equals space log space open vertical bar cose space straight pi over 4 minus cot straight pi over 4 close vertical bar space minus space log space open vertical bar cosec straight pi over 6 minus cot straight pi over 6 close vertical bar equals space log space open vertical bar square root of 2 minus 1 close vertical bar space minus space log space open vertical bar 2 minus square root of 3 close vertical bar equals space log space left parenthesis square root of 2 minus 1 right parenthesis space minus space log space left parenthesis 2 minus square root of 3 right parenthesis space equals space log open parentheses fraction numerator square root of 2 minus 1 over denominator 2 minus square root of 3 end fraction close parentheses$

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43.

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

128 Views

44.

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

93 Views

45.

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

92 Views

46.

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

99 Views

47.

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

90 Views

48.

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