Evaluate the following integral:
$space space integral subscript 0 superscript 4 fraction numerator dx over denominator square root of straight x squared plus 2 straight x plus 3 end root end fraction$

Let I = $space integral subscript 0 superscript 4 fraction numerator dx over denominator square root of straight x squared plus 2 straight x plus 3 end root end fraction space equals space integral subscript 0 superscript 4 fraction numerator 1 over denominator square root of left parenthesis straight x squared plus 2 straight x plus 1 right parenthesis plus 2 end root end fraction dx$
= $integral subscript 0 superscript 4 fraction numerator 1 over denominator square root of left parenthesis straight x plus 1 right parenthesis squared plus left parenthesis square root of 2 right parenthesis squared end root end fraction dx space equals space open square brackets log space open vertical bar left parenthesis straight x plus 1 right parenthesis plus square root of straight x squared plus 2 straight x plus 3 end root close vertical bar close square brackets subscript 0 superscript 4$
$equals space log left parenthesis 5 plus square root of 27 right parenthesis space minus space log left parenthesis 1 plus square root of 3 right parenthesis space equals space log open parentheses fraction numerator 5 plus 3 square root of 3 over denominator 1 plus square root of 3 end fraction close parentheses$

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