Question

Show that:
$integral subscript straight pi over 4 end subscript superscript straight pi over 4 end superscript open vertical bar sin space straight x close vertical bar dx space equals space 2 minus square root of 2$

Solution

Let I = $integral subscript negative straight pi over 4 end subscript superscript straight pi over 4 end superscript space open vertical bar sin space straight x close vertical bar space dx$
For $negative straight pi over 4 less or equal than straight x less or equal than 0 comma space space space open vertical bar sin space straight x close vertical bar space equals space minus sin space straight x space$
and for $0 less or equal than straight x less or equal than straight pi over 4 comma space space open vertical bar sin space straight x close vertical bar space equals space sin space straight x$
$therefore space space space straight I space equals space integral subscript negative straight pi over 4 end subscript superscript 0 open vertical bar sin space straight x close vertical bar dx plus integral subscript 0 superscript straight pi over 4 end superscript open vertical bar sin space straight x close vertical bar dx space equals space minus integral subscript negative straight pi over 4 end subscript superscript 0 sinx space dx plus integral subscript 0 superscript straight pi over 4 end superscript sinx space dx$
$equals space open square brackets cos space straight x close square brackets subscript negative straight pi over 4 end subscript superscript 0 space minus space open square brackets cos space straight x close square brackets subscript 0 superscript straight pi over 4 end superscript space equals space open square brackets cos space 0 space minus space cos space open parentheses negative straight pi over 4 close parentheses close square brackets space minus space open square brackets cos space straight pi over 4 minus cos space 0 close square brackets equals space open parentheses 1 minus fraction numerator 1 over denominator square root of 2 end fraction close parentheses space minus space open square brackets fraction numerator 1 over denominator square root of 2 end fraction minus 1 close square brackets space equals space 2 minus fraction numerator 2 over denominator square root of 2 end fraction space equals space 2 minus square root of 2$

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Integrals

Show that:
$integral subscript straight pi over 4 end subscript superscript straight pi over 4 end superscript open vertical bar sin space straight x close vertical bar dx space equals space 2 minus square root of 2$

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