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

Solution

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

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Integrals

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

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