Question

Show that:
$integral subscript negative straight pi over 4 end subscript superscript straight pi over 4 end superscript straight x cubed space space cos cubed straight x space dx space equals 0$

Solution

Let I = $integral subscript negative straight pi divided by 4 end subscript superscript straight pi divided by 4 end superscript straight x cubed space cos cubed straight x space dx$
Let f(x) = $straight x cubed space cos cubed straight x$
$therefore space space space space space straight f left parenthesis negative straight x right parenthesis space equals space left parenthesis negative straight x right parenthesis cubed space cos cubed left parenthesis negative straight x right parenthesis space equals space straight x cubed cos cubed straight x space equals space minus straight f left parenthesis straight x right parenthesis therefore space space space straight f left parenthesis straight x right parenthesis space is space an space odd space function space of space straight x therefore space space space space space straight I space equals space 0 space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space integral subscript negative straight a end subscript superscript straight a straight f left parenthesis straight x right parenthesis space dx space space equals space 0 space if space straight f left parenthesis straight x right parenthesis space is space an space odd space function close square brackets$

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Integrals

Show that:
$integral subscript negative straight pi over 4 end subscript superscript straight pi over 4 end superscript straight x cubed space space cos cubed straight x space dx space equals 0$

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