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Integrals

Short Answer Type

71.

Evaluate the following:
$integral subscript 0 superscript straight pi over 4 end superscript square root of 1 minus sin space 2 straight x end root dx$

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72.

Prove the following:
$integral subscript 0 superscript 1 open parentheses straight x space straight e to the power of straight x plus sin πx over 4 close parentheses dx space equals space 1 plus 4 over straight pi minus fraction numerator 2 square root of 2 over denominator straight pi end fraction$

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73.

Prove the following:
$integral subscript 0 superscript 1 straight x space straight e to the power of straight x space dx space equals space 1$

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74.
Evaluate
$integral subscript 0 superscript 1 open parentheses straight x space straight e to the power of straight x plus cos πx over 4 close parentheses dx$

Let I = $integral subscript 0 superscript 1 open parentheses straight x space straight e to the power of straight x plus cos πx over 4 close parentheses dx space equals integral subscript 0 superscript 1 xe to the power of straight x dx plus integral subscript 0 superscript 1 cos πx over 4 dx$
$equals left square bracket space straight x space straight e to the power of straight x right square bracket subscript 0 superscript 1 space minus space integral subscript 0 superscript 1 1. space straight e to the power of straight x dx plus open square brackets fraction numerator sin space begin display style fraction numerator straight pi space straight x over denominator 4 end fraction end style over denominator begin display style straight pi over 4 end style end fraction close square brackets subscript 0 superscript 1 space equals space left square bracket straight x space straight e to the power of straight x right square bracket subscript 0 superscript 1 minus open square brackets straight e to the power of straight x close square brackets subscript 0 superscript 1 plus 4 over straight pi open square brackets sin space πx over 4 close square brackets subscript 0 superscript 1$

$equals space left parenthesis straight e minus 0 right parenthesis space minus space left parenthesis straight e minus straight e to the power of 0 right parenthesis space plus space 4 over straight pi open parentheses sin space straight pi over 4 minus sin space 0 close parentheses$

$equals straight e minus straight e plus 1 plus 4 over straight pi open parentheses fraction numerator 1 over denominator square root of 2 end fraction minus 0 close parentheses space equals space 1 plus fraction numerator 2 square root of 2 over denominator straight pi end fraction$

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75.

Evaluate
$integral subscript 0 superscript 1 open parentheses xe to the power of 2 straight x end exponent plus sin πx over 2 close parentheses dx$

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76.

Evaluate:
$integral subscript straight pi over 2 end subscript superscript straight pi space straight e to the power of straight x open parentheses fraction numerator 1 minus sinx over denominator 1 minus cosx end fraction close parentheses dx$

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77.

If $integral subscript 0 superscript straight a 3 space straight x squared space dx space equals space 8.$
find the value of a.

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78.

If $integral subscript straight a superscript straight b straight x cubed dx space equals space 0 space and space if space integral subscript straight a superscript straight b straight x squared dx space equals space 2 over 3.$ find both a and b.

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79.

If f (x) is of the form f (x) = a+b x+cx², show that
$integral subscript 0 superscript 1 straight f left parenthesis straight x right parenthesis space dx space equals space 1 over 6 open curly brackets straight f left parenthesis 0 right parenthesis space plus space 4 space straight f space open parentheses 1 half close parentheses plus straight f left parenthesis 1 right parenthesis close curly brackets.$

148 Views

80.

Prove the following:
$integral subscript 0 superscript 1 sin to the power of negative 1 end exponent straight x space dx space equals space straight pi over 2 minus 1$

104 Views

Previous Year Papers

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Test Yourself

Short Answer Type

74. Evaluate

74.
Evaluate
$integral subscript 0 superscript 1 open parentheses straight x space straight e to the power of straight x plus cos πx over 4 close parentheses dx$