I = ∫ex(x2-1 + 2(x+1)2) dx = ∫ex(x-1x+1 + 2(x+1)2) dx = ex(x-1x+1) +c

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$\int e^{x} (\frac{x^{2} + 1}{{(x + 1)}^{2}}) dx =$

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$e^{x} (\frac{x-1}{x+1}) + c$

$e^{- x} (\frac{x+1}{x -1}) + c$

$e^{x} (\frac{x+1}{x^{2} +1}) + c$

$e^{x} (\frac{1}{x +1}) + c$

answer is A.

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Detailed Solution

$I = \int e^{x} (\frac{x^{2} -1 + 2}{{(x+1)}^{2}}) dx$

$= \int e^{x} (\frac{x-1}{x+1} + \frac{2}{{(x+1)}^{2}}) dx$

${= e}^{x} (\frac{x-1}{x+1}) +c$

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