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71.
Evaluate the following:
$integral subscript 0 superscript straight pi over 4 end superscript square root of 1 minus sin space 2 straight x end root dx$

Let I = $integral subscript 0 superscript straight pi over 4 end superscript square root of 1 minus sin space 2 straight x end root dx space equals space integral subscript 0 superscript straight pi over 4 end superscript square root of cos squared straight x plus sin squared straight x minus 2 sinx space cosx end root dx$
$equals space integral subscript 0 superscript straight pi over 4 end superscript space square root of left parenthesis cosx minus sinx right parenthesis squared end root space dx space equals space integral subscript 0 superscript straight pi over 4 end superscript left parenthesis cosx minus sinx right parenthesis space dx$
$equals left square bracket sinx plus cosx right square bracket subscript 0 superscript straight pi over 4 end superscript space equals space open parentheses sin straight pi over 4 plus cos straight pi over 4 close parentheses minus left parenthesis sin 0 plus cos 0 right parenthesis equals space open parentheses fraction numerator 1 over denominator square root of 2 end fraction plus fraction numerator 1 over denominator square root of 2 end fraction close parentheses space minus space left parenthesis 0 plus 1 right parenthesis space equals space fraction numerator 2 over denominator square root of 2 end fraction minus 1 space equals square root of 2 minus 1.$

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

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80.

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