Evaluate $integral subscript negative 1 end subscript superscript 1 sin to the power of 5 straight x space cos to the power of 4 straight x space dx$

Solution

Comparing $integral subscript negative 1 end subscript superscript 1 space sin to the power of 5 straight x space cos to the power of 4 straight x space dx space with space integral subscript negative 1 end subscript superscript 1 space straight f left parenthesis straight x right parenthesis space dx comma space space we space get$
f (x) = sin⁵ x cos⁴ x
Now f (–x) = sin⁵ (–x) cos⁴ (–x) = – sin⁵x cos⁴ x = – f (x)
∴ f (x) is an odd function
$therefore space space space space straight I space equals space 0$ [ $integral subscript negative straight a end subscript superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space 0 space space if space straight f left parenthesis straight x right parenthesis$ is an odd function]
$therefore space space space integral subscript negative 1 end subscript superscript 1 space sin to the power of 5 straight x space cos to the power of 4 straight x space dx space equals space 0$

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Integrals

Evaluate $integral subscript negative 1 end subscript superscript 1 sin to the power of 5 straight x space cos to the power of 4 straight x space dx$

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