Question

Evaluate $integral subscript 0 superscript straight pi space sin squared straight x space cos cubed straight x space dx.$

Solution

Let I = $integral subscript 0 superscript straight pi sin squared straight x space cos cubed straight x space dx space equals space integral subscript 0 superscript straight pi straight f left parenthesis straight x right parenthesis space dx$
where f(x) = $sin squared straight x space cos cubed straight x$
Now $straight f left parenthesis straight pi minus straight x right parenthesis space equals space sin squared left parenthesis straight pi minus straight x right parenthesis space cos cubed left parenthesis straight pi minus straight x right parenthesis space equals space minus sin squared straight x space cos cubed straight x$
$therefore space space space straight f left parenthesis straight pi minus straight x right parenthesis space space equals negative straight f left parenthesis straight x right parenthesis$
$therefore space space space integral subscript 0 superscript straight pi sin squared straight x space cosx space dx space equals space 0 space space space space space open square brackets because space space if space straight f left parenthesis 2 straight a minus straight x right parenthesis space equals space minus straight f left parenthesis straight x right parenthesis comma space then space integral subscript straight a superscript 2 straight a end superscript straight f left parenthesis straight x right parenthesis space dx space equals space 0 close square brackets$

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Integrals

Evaluate $integral subscript 0 superscript straight pi space sin squared straight x space cos cubed straight x space dx.$

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