Question

Evaluate $integral subscript negative 1 end subscript superscript 1 space straight x to the power of 17 space cos to the power of 4 straight x space dx.$

Solution

Comparing $integral subscript negative 1 end subscript superscript 1 straight x to the power of 17 space cos to the power of 4 straight x space dx space with space integral subscript negative 1 end subscript superscript 1 straight f left parenthesis straight x right parenthesis space dx comma space we space get comma$
$straight f left parenthesis straight x right parenthesis space equals space straight x to the power of 17 space cos to the power of 4 straight x$
Now $straight f left parenthesis negative straight x right parenthesis space equals space left parenthesis negative straight x right parenthesis to the power of 17 space cos to the power of 4 colon left parenthesis negative straight x right parenthesis space equals space minus straight x to the power of 17 space cos to the power of 4 straight x space equals space space minus straight f left parenthesis straight x right parenthesis$
$therefore space space space space straight f left parenthesis straight x right parenthesis space is space an space odd space function$
$therefore space space space space space straight I space equals space 0 space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space integral subscript negative straight a end subscript superscript straight a straight f vertical line left parenthesis straight x right parenthesis space dx space equals space 0 space if space straight f space is space an space odd space function close square brackets$
$therefore space space space space integral subscript negative 1 end subscript superscript 1 space straight x to the power of 17 space cos to the power of 4 straight x space dx space equals space 0$

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Integrals

Evaluate $integral subscript negative 1 end subscript superscript 1 space straight x to the power of 17 space cos to the power of 4 straight x space dx.$

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