Evaluate the following integral
![\[ \int \frac{e^{\arctan x}}{(1+x^2)^{\frac{3}{2}}} \, dx. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-92232c7597b6d6415a606cb054ab7407_l3.png "Rendered by QuickLaTeX.com")
From the previous exercise (Section 6.22, Exercise #40) we know
![\[ \int \frac{e^{arctan x}}{(1+x^2)^{\frac{3}{2}}} \, dx = \int \frac{x e^{\arctan x}}{(1+x^2)^{\frac{3}{2}}} \, dx + \frac{e^{\arctan x}}{\sqrt{1+x^2}}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-6e94b18852954bf49f1e0ee3440e0c24_l3.png "Rendered by QuickLaTeX.com")
From the same exercise we also know that
![\[ \int \frac{x e^{\arctan x}}{(1+x^2)^{\frac{3}{2}}} \, dx = \frac{(x-1)e^{\arctan x}}{2 \sqrt{1+x^2}}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-1d5d32fb73dd545ff458fc93aea902f8_l3.png "Rendered by QuickLaTeX.com")
Therefore, we have
![\begin{align*} \int \frac{e^{\arctan x}}{(1+x^2)^{\frac{3}{2}}} \, dx &= \frac{(x-1)e^{\arctan x} + 2e^{\arctan x}}{2 \sqrt{1+x^2}} + C \\[10pt] &= \frac{(x+1)e^{\arctan x}}{2\sqrt{1+x^2}} + C. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-a5770e41344685d07f4f4c35cf74626f_l3.png "Rendered by QuickLaTeX.com")
Evaluate the following integral
From the previous exercise (Section 6.22, Exercise #40) we know
From the same exercise we also know that
Therefore, we have