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

115 Views

76.

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

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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.

integral subscript straight a superscript straight b space straight x cubed space dx space equals space 0 space space space rightwards double arrow space space open square brackets straight x to the power of 4 over 4 close square brackets subscript straight a superscript straight b space equals space 0

rightwards double arrow space space space space 1 fourth left parenthesis straight b to the power of 4 minus straight a to the power of 4 right parenthesis space space equals space 0 space rightwards double arrow space straight b to the power of 4 minus straight a to the power of 4 space equals space 0 rightwards double arrow space space space space space left parenthesis straight b squared minus straight a squared right parenthesis space left parenthesis straight b squared plus straight a squared right parenthesis space equals space 0 space space rightwards double arrow space space straight b squared minus straight a squared space equals space 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space straight b squared plus straight a squared space not equal to 0 close square brackets rightwards double arrow space space space space space left parenthesis straight b minus straight a right parenthesis space left parenthesis straight b plus straight a right parenthesis space equals space 0 space space rightwards double arrow space space space straight b plus straight a space equals space 0 space space space space space space space open square brackets because space straight b not equal to straight a space as space integral subscript straight a superscript straight b straight x squared dx not equal to 0 close square brackets therefore space space space space straight b space equals space minus straight a space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis

Also

integral subscript straight a superscript straight b straight x squared dx space equals space 2 over 3 space space space space space space space space rightwards double arrow space space space space space space space open square brackets straight x cubed over 3 close square brackets subscript straight a superscript straight b space equals space 2 over 3

rightwards double arrow

1 third left parenthesis straight b cubed minus straight a cubed right parenthesis space equals space 2 over 3

rightwards double arrow

straight b cubed minus straight a cubed space equals space 2

therefore

left parenthesis negative straight a right parenthesis cubed minus straight a cubed space equals space 2

open square brackets because space space of space left parenthesis 1 right parenthesis close square brackets

therefore

negative space 2 space straight a cubed space equals space 2

rightwards double arrow

straight a cubed equals negative 1

rightwards double arrow

a = -1

therefore

we have a = -1, b = 1

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

78. If find both a and b.

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.