Evaluate $integral subscript negative straight pi over 4 end subscript superscript straight pi over 4 end superscript cos squared xdx$

Solution

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

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Integrals

Evaluate $integral subscript negative straight pi over 4 end subscript superscript straight pi over 4 end superscript cos squared xdx$

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