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

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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
$integral subscript 0 superscript 1 straight f left parenthesis straight x right parenthesis space dx space equals space 1 over 6 open curly brackets straight f left parenthesis 0 right parenthesis space plus space 4 space straight f space open parentheses 1 half close parentheses plus straight f left parenthesis 1 right parenthesis close curly brackets.$

Here $straight f left parenthesis straight x right parenthesis space equals space straight a plus straight b plus cx squared$
$therefore space space straight f left parenthesis 0 right parenthesis space equals space straight a plus 0 plus 0 space equals space straight a comma space space space straight f open parentheses 1 half close parentheses space equals space straight a plus straight b over straight c plus straight c over 4 comma space space space straight f left parenthesis 1 right parenthesis space equals space straight a plus straight b plus straight c$
R.H.S. = $1 over 6 open curly brackets straight f left parenthesis 0 right parenthesis plus 4 space straight f open parentheses 1 half close parentheses plus straight f left parenthesis 1 right parenthesis close curly brackets space equals space 1 over 6 open curly brackets straight a plus 4 open parentheses straight a plus straight b over 2 plus straight c over 4 close parentheses plus left parenthesis straight a plus straight b plus straight c right parenthesis close curly brackets$

$equals 1 over 6 left curly bracket straight a plus 4 straight a plus 2 straight b plus straight c plus straight a plus straight b plus straight c right curly bracket space equals space 1 over 6 left parenthesis 6 straight a plus 3 straight b plus 2 straight c right parenthesis space equals space straight a plus straight b over 2 plus straight c over 3$
$straight L. straight H. straight S. space equals space integral subscript 0 superscript 1 straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 0 superscript 1 left parenthesis straight a plus bx plus cx squared right parenthesis space dx space equals space open square brackets ax plus bx squared over 2 plus cx cubed over 3 close square brackets subscript 0 superscript 1$
$equals space open parentheses straight a plus straight b over 2 plus straight c over 3 close parentheses space minus space left parenthesis 0 plus 0 plus 0 right parenthesis space equals space straight a plus straight b over 2 plus straight c over 3$
∴ L.H.S. = R.H.S. Hence the result.

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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Short Answer Type

79. If f (x) is of the form f (x) = a+b x+cx2, show that