**Intuitively, the result is obvious** because (a) $\phi$ is a [rapidly decreasing function](https://en.wikipedia.org/wiki/Schwartz_space#Examples_of_functions_in_the_Schwartz_space) (its magnitude decreases at a quadratic exponential rate) and (b) $\Phi$ is bounded above and, for negative $x,$ is also rapidly decreasing at essentially the same rate as $\phi.$  Thus the fraction $\phi^2/\Phi$ decreases rapidly for both positive and negative $x,$ while remaining bounded in between, whence its integral is *very* nicely behaved and finite.

[![Figure][1]][1]

**The problem, then, is to make this intuition rigorous.**  The rigor merely parallels the foregoing argument by making suitable quantitative comparisons.


When $x\gt 0,$ $\Phi(x)\ge 1/2$ (by a familiar symmetry argument).  Whence (in this case) the integrand is bounded above in magnitude by

$$\bigg|\frac{\phi(x)^2}{\Phi(x)}\bigg| \le 2\phi(x)^2 \lt 2\exp(-x^2)/\sqrt{2\pi}$$

which (because it is proportional to the density of another Normal distribution) has a finite integral.

The harder part is the integral over negative $x.$  But here, the [Mills Ratio](https://stats.stackexchange.com/a/7206/919) is

$$R(-x) = \frac{\Phi(x)}{\phi(x)}$$

which, as the linked post explains, is bounded below by $-x/(x^2+1).$  Thus, for large negative $x$ (say, $x \le -1$),

$$\bigg|\frac{\phi(x)^2}{\Phi(x)}\bigg| = \bigg|\phi(x)\left(\frac{1}{R(-x)}\right)\bigg| \le \phi(x) \frac{x^2+1}{|x|} \le 2|x|\phi(x)$$

whose integral also converges (it can integrated exactly using elementary techniques).
[![Figure 2][2]][2]

Since $\phi(x)^2/\Phi(x)$ is bounded on the remaining interval $[-1,0],$ we have established that its integral over the entire real line is bounded in magnitude by the sum of three convergent quantities, whence it is finite, *QED.*

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This argument continues to apply, essentially without change, to any integrand of the form $\phi(x)^k/\Phi(x)$ for $k\gt 1.$  It also shows (look more closely at the upper and lower bounds for Mills' Ratio) that the integral *diverges* when $k\le 1.$


  [1]: https://i.sstatic.net/zx5nj.png
  [2]: https://i.sstatic.net/0oM5A.png