Question

Show that:
$integral subscript negative straight pi over 2 end subscript superscript straight pi over 4 end superscript space sin to the power of 7 straight x space dx space equals space 0$

Solution

Let I = $integral subscript negative straight pi divided by 2 end subscript superscript straight pi divided by 2 end superscript space sin to the power of 7 straight x space dx$
Comparing $integral subscript negative straight pi divided by 2 end subscript superscript straight pi divided by 2 end superscript space sin to the power of 7 space straight x space dx space space with space integral subscript negative straight pi divided by 2 end subscript superscript straight pi divided by 2 end superscript space straight f left parenthesis straight x right parenthesis space dx comma space we space get$
$straight f left parenthesis straight x right parenthesis space equals space sin to the power of 7 straight x$
Now $straight f left parenthesis negative straight x right parenthesis space equals space sin to the power of 7 left parenthesis negative straight x right parenthesis space equals space minus sin to the power of 7 straight x space equals space minus straight f left parenthesis straight x right parenthesis$
$therefore space space space straight f left parenthesis straight x right parenthesis space is space an space odd space function$
$therefore space space space straight I space equals space 0 space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space integral subscript negative straight a end subscript superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space 0 space if space straight f left parenthesis straight x right parenthesis space is space an space odd space function close square brackets$

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Integrals

Show that:
$integral subscript negative straight pi over 2 end subscript superscript straight pi over 4 end superscript space sin to the power of 7 straight x space dx space equals space 0$

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