Solution
= Solution
Split the <integral> into
$$
\int\frac{dx}{\sqrt{1+x^2}}+\int\frac{x\,dx}{\sqrt{1+x^2}}.
$$
The first is the standard <antiderivative> $\operatorname{arsinh}x=\ln(x+\sqrt{1+x^2})$ from the <inverse hyperbolic functions>. For the second, the <substitution rule> $t=1+x^2$ gives $\tfrac12\int t^{-1/2}\,dt=\sqrt{1+x^2}$. Hence
$$
\boxed{\ln(x+\sqrt{1+x^2})+\sqrt{1+x^2}+C.}
$$
The <logarithm>'s argument is positive for every real $x$.