Evaluate $integral subscript negative straight pi over 4 end subscript superscript straight pi over 4 end superscript space sin squared straight x space dx$

Solution

Let I = $integral subscript straight pi over 4 end subscript superscript straight pi over 4 end superscript sin squared straight x space dx space equals space 2 integral subscript 0 superscript straight pi over 4 end superscript sin squared straight x space dx$
[∵ sin² x is an even function as sin² (–x) = sin² x]
$equals space integral subscript 0 superscript straight pi over 4 end superscript space space 2 space sin squared straight x space dx space equals space integral subscript 0 superscript straight pi over 4 end superscript left parenthesis 1 minus cos space 2 straight x right parenthesis space dx space equals space open square brackets straight x minus fraction numerator sin space 2 straight x over denominator 2 end fraction close square brackets subscript 0 superscript straight pi over 4 end superscript equals space open parentheses straight pi over 4 minus 1 half sin straight pi over 2 close parentheses space minus space open parentheses 0 minus 1 half sin space 0 close parentheses space equals space open parentheses straight pi over 4 minus 1 half close parentheses space minus space space left parenthesis 0 space minus space 0 right parenthesis space equals space straight pi over 4 minus 1 half$

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Integrals

Evaluate $integral subscript negative straight pi over 4 end subscript superscript straight pi over 4 end superscript space sin squared straight x space dx$

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