<s>[INST] How to distribute $x \times \sqrt{x^2 + 1}$ How do I distribute the x in this problem?
How do I "gain access" so to say. Does it become $x^\frac{1}{2}(x^2 + 1)^\frac{1}{2}$ or $(x^3+x)^\frac{1}{2}$ ?
Or do I need to even do that in order to integrate $\int x(x^2+1)^\frac{1}{2}$ ?
 [/INST] You don't need to expand in order to integrate. Just use a $u$ substitution with $u = x^2 + 1$ and so $du = 2x \space dx$ and so $dx = \frac{du}{2x}$ and so
your integral becomes $$\large{\int \color{red}{x}u^\frac{1}{2} \times \frac{du}{2\color{red}{x}}}$$ and so you cancel out the $x$ to end up with $$ \large{\int u^\frac{1}{2} du = \frac{u^\frac{3}{2}}{ \frac{3}{2}}} =\frac {2u^\frac{3}{2}}{3}=\frac{2(x^2+1)^\frac{3}{2}}{3} + C$$
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