Show that:
$integral subscript 0 superscript straight pi over 2 end superscript space open vertical bar sin space straight x space cosx close vertical bar space dx space equals space 1 half$

Solution

Let I = $integral subscript 0 superscript straight pi over 2 end superscript open vertical bar sinx space cosx close vertical bar dx space equals space 1 half integral subscript 0 superscript straight pi over 2 end superscript open vertical bar 2 space sinx space cosx close vertical bar dx$
$equals space 1 half integral subscript 0 superscript straight pi over 2 end superscript open vertical bar sin space 2 straight x close vertical bar dx space equals space 1 half integral subscript 0 superscript straight pi over 2 end superscript sin space 2 straight x space dx space equals space 1 half open square brackets fraction numerator negative cos space 2 straight x over denominator 2 end fraction close square brackets subscript 0 superscript straight pi over 2 end superscript space equals space minus 1 fourth open square brackets cos space 2 straight x close square brackets subscript 0 superscript straight pi over 2 end superscript$
$equals negative 1 fourth left square bracket cos space straight pi space minus space cos space 0 right square bracket space equals space minus 1 fourth left square bracket negative 1 minus 1 right square bracket space space equals space 1 half$

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Integrals

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript space open vertical bar sin space straight x space cosx close vertical bar space dx space equals space 1 half$

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