<s>[INST] Improper integral that converges for all $x$ in $ \mathbb{R}$ 
Let $f(x)$ be defined by the improper integral:
  $$f(x)= \int_{0}^{\infty} \cos\left(\frac{t^3}{3} + \frac{x^2 t^2}{2} + xt\right)dt.$$ 
  Show that this improper integral converges for all $x \in\mathbb{R}$.

How do you start this question? Do I evaluate the integral first, and then see if it satisfies all $x \in\mathbb{R}$?
 [/INST] Let $x\in\mathbb R$, then there exists an $N>0$ such that $t^2+x^2t+x>0$ for all $t>N$. Then, if $M>N$: you have that $$\int_N^M\cos\left(\frac{t^3}{3}+\frac{x^2t^2}{2}+xt\right)\,dt=\int_N^M\left(\sin\left(\frac{t^3}{3}+\frac{x^2t^2}{2}+xt\right)\right)'\frac{1}{t^2+x^2t+x}\,dt=\left[\sin\left(\frac{t^3}{3}+\frac{x^2t^2}{2}+xt\right)\frac{1}{t^2+x^2t+x}\right]_N^M+\int_N^M\sin\left(\frac{t^3}{3}+\frac{x^2t^2}{2}+xt\right)\frac{2t+x^2}{(t^2+x^2t+x)^2}\,dt.$$ The first term converges as $M\to\infty$, and the last integrand is bounded by $$\left|\frac{2t+x^2}{(t^2+x^2t+x)^2}\right|\leq \frac{ct}{t^4}=\frac{c}{t^3},$$ where $c$ is a positive constant, so it converges as $M\to\infty$.
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