∫ex11+x2+1−2×21+x25dx

$\int {\mathrm{e}}^{\mathrm{x}}\left[\frac{1}{\sqrt{1+{\mathrm{x}}^{2}}}+\frac{1-2{\mathrm{x}}^{2}}{\sqrt{{\left(1+{\mathrm{x}}^{2}\right)}^{5}}}\right]\mathrm{dx}$

1. A

${\mathrm{e}}^{\mathrm{x}}\left[\frac{1}{\sqrt{1+{\mathrm{x}}^{2}}}+\frac{\mathrm{x}}{\sqrt{{\left(1+{\mathrm{x}}^{2}\right)}^{3}}}\right]+\mathrm{C}$

2. B

${\mathrm{e}}^{\mathrm{x}}\left[\frac{1}{\sqrt{1+{\mathrm{x}}^{2}}}-\frac{\mathrm{x}}{\sqrt{{\left(1+{\mathrm{x}}^{2}\right)}^{3}}}\right]+\mathrm{C}$

3. C

${\mathrm{e}}^{\mathrm{x}}\left[\frac{1}{\sqrt{1+{\mathrm{x}}^{2}}}+\frac{\mathrm{x}}{\sqrt{{\left(1+{\mathrm{x}}^{2}\right)}^{5}}}\right]+\mathrm{C}$

4. D

${\mathrm{e}}^{\mathrm{x}}\left[\frac{1}{\sqrt{1-{\mathrm{x}}^{2}}}-\frac{\mathrm{x}}{\sqrt{\left(1+{\mathrm{x}}^{2}\right)}}\right]+\mathrm{C}$

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$\begin{array}{l}I=\int {e}^{x}\left[\frac{1}{\sqrt{1+{x}^{2}}}-\frac{x}{\sqrt{{\left(1+{x}^{2}\right)}^{3}}}+\frac{x}{\sqrt{{\left(1+{x}^{2}\right)}^{3}}}+\frac{1-2{x}^{2}}{\sqrt{{\left(1+{x}^{2}\right)}^{5}}}\right]dx\\ I={e}^{x}\frac{1}{\sqrt{1+{x}^{2}}}+{e}^{x}\frac{x}{\sqrt{{\left(1+{x}^{2}\right)}^{3}}}+C\\ I={e}^{x}\left[\frac{1}{\sqrt{1+{x}^{2}}}+\frac{x}{\sqrt{{\left(1+{x}^{2}\right)}^{3}}}\right]+C\end{array}$