91Ó°ÊÓ

First, notice that \(\sec^2 x = 1 + \tan^2 x\). So, we can rewrite the given integral as: $$\int x^2 \sec^2 x dx = \int x^2 (1 + \tan^2 x) dx$$ Now, we can split the integral into two integrals: $$\int x^2 (1 + \tan^2 x) dx = \int x^2 dx + \int x^2 \tan^2 x dx$$ Since \(\int x^2 dx\) is elementary (\(\frac{1}{3}x^3\)), we only need to show that \(\int x^2 \tan^2 x dx\) is not elementary, given that \(\int x \tan x dx\) is not elementary.

Let's take derivative of \(x\tan x\) with respect to \(x\). Using product rule, we get: $$\frac{d}{dx}(x\tan x) = x \sec^2 x + \tan x $$ Since \(\int x\tan x dx\) is not elementary, its antiderivative cannot be expressed as a combination of elementary functions. Therefore, if the integral \(\int x^2 \tan^2 x dx\) were elementary, we would have an elementary function whose derivative is \(x\tan x\). However, we know that this is not possible, so \(\int x^2 \tan^2 x dx\) is not elementary either.

In order to analyze this integral, let's simplify its integrand by using the identity \(\cos x = \frac{1 - \tan^2(\frac{x}{2})}{1 + \tan^2(\frac{x}{2})}\). This identity will help us to rewrite the integral in a form that will allow us to determine whether it is elementary or not. So, we have: $$\frac{x^2}{1+\cos x} = x^2\frac{1+\tan^2(\frac{x}{2})}{2(1+\tan^2(\frac{x}{2}))}$$ Now, we can split this expression into two integrals: $$\int x^2\frac{1+\tan^2(\frac{x}{2})}{2(1+\tan^2(\frac{x}{2}))} dx = \frac{1}{2} \int x^2 dx + \frac{1}{2} \int x^2 \tan^2(\frac{x}{2}) dx$$ Since \(\int x^2 dx\) is elementary (as we discussed before), we only need to show that \(\int x^2 \tan^2(\frac{x}{2}) dx\) is not elementary, given that \(\int x \tan x dx\) is not elementary. Consider the substitution \(u = \frac{x}{2}\), then we have \(x = 2u\) and \(dx = 2 du\). So, the integral becomes: $$\int x^2 \tan^2(\frac{x}{2}) dx = \int (2u)^2 \tan^2(u)(2 du) = 8\int u^2 \tan^2(u) du$$ Since \(\int u^2 \tan^2(u) du\) is not elementary (from (B)), the integral of the original function \(\int \frac{x^2 dx}{1+\cos x}\) is also not elementary. So, we have deduced that (A) \(\int x^2 \sec^2 x dx\), (B) \(\int x^2 \tan^2 x dx\), and (C) \(\int \frac{x^2 dx}{1+\cos x}\) are not elementary given the fact that \(\int x\tan x dx\) is not elementary.