Question
Evalute : $\int \sqrt{x^2+2 x+5} d x$

Answer

$
\begin{aligned}
& \int \sqrt{x^2+2 x+5} d x \\
= & \int \sqrt{\left(x^2+2 x+1\right)+4} d x \\
= & \int \sqrt{(x+1)^2+(2)^2} d x \\
= & \frac{(x+1)}{2} \sqrt{(x+1)^2+(2)^2}+
\end{aligned}
$
$
\frac{(2)^2}{2} \log \left|(x+1)+\sqrt{(x+1)^2+(2)^2}\right|+c
$
$
=\frac{(x+1)}{2} \sqrt{x^2+2 x+5}+
$
$
2 \log \left|(x+1)+\sqrt{x^2+2 x+5}\right|+c .
$

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