Exercise 8

Consider the indefinite integral given by $$\begin{gathered} \int \frac{\sqrt{x+\sqrt{1+x^2}}}{x} \, dx \end{gathered}$$.

1. Explain why $u$-substitution does not offer a way to simplify this integral by discussing at least two different options you might try for $u$.

2. Explain why integration by parts does not seem to be a reasonable way to proceed, either, by considering one option for $u$ and $dv$.

3. Is there any line in the integral table in Appendix that is helpful for this integral?

4. Evaluate the given integral using WolframAlpha. What do you observe?