Prove the following:
$integral subscript 0 superscript 1 open parentheses straight x space straight e to the power of straight x plus sin πx over 4 close parentheses dx space equals space 1 plus 4 over straight pi minus fraction numerator 2 square root of 2 over denominator straight pi end fraction$

Solution

Let I = $integral subscript 0 superscript 1 open parentheses xe to the power of straight x plus sin πx over 4 close parentheses space dx$
$equals space integral subscript 0 superscript 1 straight x space straight e to the power of straight x space dx space plus space integral subscript 0 superscript 1 sin πx over 4 dx space equals space left square bracket straight x space straight e to the power of straight x right square bracket subscript 0 superscript 1 space minus space integral subscript 0 superscript 1 1. space straight e to the power of straight x dx plus open square brackets fraction numerator negative cos begin display style πx over 4 end style over denominator begin display style straight pi over 4 end style end fraction close square brackets subscript 0 superscript 1$
$equals left square bracket straight x space straight e to the power of straight x right square bracket subscript 0 superscript 1 space minus space left square bracket straight e to the power of straight x right square bracket subscript 0 superscript 1 space minus space 4 over straight pi open square brackets cos fraction numerator straight pi space straight x over denominator 4 end fraction close square brackets subscript 0 superscript 1 space equals space left parenthesis straight e minus 0 right parenthesis minus left parenthesis straight e minus straight e to the power of 0 right parenthesis minus 4 over straight pi open square brackets cos straight pi over 4 minus cos space 0 close square brackets equals space straight e minus straight e plus 1 minus 4 over straight pi open parentheses fraction numerator 1 over denominator square root of 2 end fraction minus 1 close parentheses space equals space 1 plus 4 over straight pi minus fraction numerator 2 square root of 2 over denominator straight pi end fraction$

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