Evaluate the following:
$integral subscript 0 superscript 2 straight pi end superscript straight e to the power of straight x cos open parentheses straight pi over 4 plus straight x over 2 close parentheses dx.$

Solution

Let
I = $integral subscript 0 superscript 2 straight pi end superscript straight e to the power of straight x space cos open parentheses straight pi over 4 plus straight x over 2 close parentheses dx$
Put $straight pi over 4 plus straight x over 2 space equals space straight t comma space space space space space space space therefore space space 1 half dx space equals space dt space space space space space space space space rightwards double arrow space space space dx space equals space 2 space dt$
When $straight x space equals space 0 comma space space space straight t space equals space straight pi over 4$
When    $straight x space equals space 2 straight pi comma space space space straight t space equals space fraction numerator 5 straight pi over denominator 4 end fraction$
$therefore$ $straight I space equals space 2 integral subscript straight pi over 4 end subscript superscript fraction numerator 5 straight pi over denominator 4 end fraction end superscript straight e to the power of 2 straight t minus straight pi over 2 end exponent space cos space straight t space dt space equals space 2 space straight e to the power of negative straight pi over 2 end exponent integral subscript straight pi over 4 end subscript superscript fraction numerator 5 straight pi over denominator 4 end fraction end superscript straight e to the power of 2 straight t end exponent space cost space dt space$
   $equals space 2 space straight e to the power of negative straight pi over 2 end exponent. space fraction numerator 1 over denominator 4 plus 1 end fraction open square brackets straight e to the power of 2 straight t end exponent space left parenthesis 2 space cos space straight t space plus space sin space straight t close square brackets subscript straight pi over 4 end subscript superscript fraction numerator 5 straight pi over denominator 4 end fraction end superscript$
   $equals space 2 over 5 straight e to the power of negative straight pi over 2 end exponent open square brackets straight e to the power of fraction numerator 5 straight pi over denominator 2 end fraction end exponent open parentheses 2 space cos space fraction numerator 5 straight pi over denominator 4 end fraction plus sin fraction numerator 5 straight pi over denominator 4 end fraction close parentheses space minus space straight e to the power of straight pi over 2 end exponent space open parentheses 2 space cos space straight pi over 4 plus sin straight pi over 4 close parentheses close square brackets$
$equals space 2 over 5 straight e to the power of negative straight pi over 2 end exponent space open square brackets straight e to the power of fraction numerator 5 straight pi over denominator 2 end fraction end exponent open parentheses negative fraction numerator 2 over denominator square root of 2 end fraction minus fraction numerator 1 over denominator square root of 2 end fraction close parentheses minus straight e to the power of straight pi over 2 end exponent open parentheses fraction numerator 2 over denominator square root of 2 end fraction plus fraction numerator 1 over denominator square root of 2 end fraction close parentheses close square brackets equals space 2 over 5 straight e to the power of negative straight pi over 2 end exponent open square brackets negative fraction numerator 3 over denominator square root of 2 end fraction straight e to the power of fraction numerator 5 straight pi over denominator 2 end fraction end exponent minus fraction numerator 3 over denominator square root of 2 end fraction straight e to the power of straight pi over 2 end exponent close square brackets space equals negative fraction numerator 3 square root of 2 over denominator 5 end fraction left parenthesis straight e to the power of 2 straight pi end exponent plus 1 right parenthesis$

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Integrals

Evaluate the following:
$integral subscript 0 superscript 2 straight pi end superscript straight e to the power of straight x cos open parentheses straight pi over 4 plus straight x over 2 close parentheses dx.$

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