Can use FToC to evaluate lim x → ∞ ∫ 0 x f ( t...

Because $f(x)\ge \frac{1}{2}{x}^2-2$, $\underset{x\to \mathrm{\infty }}{lim}f(x)=\mathrm{\infty }$. Hence the limit is of the indeterminate form $\frac{\mathrm{\infty }}{\mathrm{\infty }}$ and we apply L'Hopital's rule. We find the derivative of the numerator by the FTC, getting
$\underset{x\to \mathrm{\infty }}{lim}\frac{{\int }_0^{x}f(t)dt}{x^2}=\underset{x\to \mathrm{\infty }}{lim}\frac{f(x)}{2x}=\underset{x\to \mathrm{\infty }}{lim}\frac{\cos x}{2x}-\frac{1}{2x}+\frac{x}{4}$
These three limits evaluate to $0$ (by the squeeze theorem), $0$, and $+\mathrm{\infty }$, respectively; hence their sum has limit $+\mathrm{\infty }$.