<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] Note that 
$$\sqrt{x^2}=|x|.$$
If $x\ge 0$, then $x=|x|=\sqrt{x^2}$ leads $$x\sqrt{x^2+1}=\sqrt{x^2}\cdot\sqrt{x^2+1}=\sqrt{x^2(x^2+1)}.$$
If $x\lt 0$, then $x=-|x|=-\sqrt{x^2}$ leads$$x\sqrt{x^2+1}=-\sqrt{x^2}\cdot\sqrt{x^2+1}=-\sqrt{x^2(x^2+1)}.$$
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