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

109 Views

74.

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

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75.

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

115 Views

76.
Evaluate:
$integral subscript straight pi over 2 end subscript superscript straight pi space straight e to the power of straight x open parentheses fraction numerator 1 minus sinx over denominator 1 minus cosx end fraction close parentheses dx$

Let I = $integral subscript straight pi over 2 end subscript superscript straight pi space straight e to the power of straight x open parentheses fraction numerator 1 minus sinx over denominator 1 minus cosx end fraction close parentheses dx space equals space integral subscript straight pi over 2 end subscript superscript straight pi straight e to the power of straight x space open square brackets fraction numerator 1 minus 2 sin begin display style straight x over 2 end style cos begin display style straight x over 2 end style over denominator 2 sin squared begin display style straight x over 2 end style end fraction close square brackets dx$

$equals space integral subscript straight pi over 2 end subscript superscript straight pi space straight e to the power of straight x open square brackets fraction numerator 1 over denominator 2 sin squared begin display style straight x over 2 end style end fraction minus fraction numerator 2 sin begin display style straight x over 2 end style cos begin display style straight x over 2 end style over denominator 2 sin squared begin display style straight x over 2 end style end fraction close square brackets space dx space equals space integral subscript straight pi over 2 end subscript superscript straight pi straight e to the power of straight x open square brackets 1 half cosec squared straight x over 2 minus cot straight x over 2 close square brackets dx$

$equals space integral subscript straight pi over 2 end subscript superscript straight pi over 2 end superscript straight e to the power of straight x space open square brackets negative cot straight x over 2 plus 1 half cosec squared straight x over 2 close square brackets dx$

$equals space open square brackets straight e to the power of straight x space open parentheses negative cot straight x over 2 close parentheses close square brackets subscript straight pi over 2 end subscript superscript straight pi$ $open square brackets because space space space integral space straight e to the power of straight x space left curly bracket straight f left parenthesis straight x right parenthesis plus straight f apostrophe left parenthesis straight x right parenthesis right curly bracket dx space equals space straight e to the power of straight x space straight f left parenthesis straight x right parenthesis close square brackets$

$equals negative open square brackets straight e to the power of straight x open parentheses negative cot straight x over 2 close parentheses close square brackets subscript straight pi over 2 end subscript superscript straight pi space equals space minus open square brackets straight e to the power of straight pi cot straight pi over 2 minus straight e to the power of straight pi over 2 end exponent cot straight pi over 4 close square brackets equals negative open square brackets straight e to the power of straight pi space cross times 0 minus straight e to the power of straight pi over 2 end exponent cross times 1 close square brackets space equals space straight e to the power of straight pi over 2 end exponent$

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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.

105 Views

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.

101 Views

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

76. Evaluate:

76.
Evaluate:
$integral subscript straight pi over 2 end subscript superscript straight pi space straight e to the power of straight x open parentheses fraction numerator 1 minus sinx over denominator 1 minus cosx end fraction close parentheses dx$