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 $x \in\mathbb{R}$?
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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 $M\to\infty$.
I think the general idea would be to notice that for all $x$ you can find $A,s$ such that: $$A t^2 < \frac{t^3}{3} + \frac{t^2 x^2}{2} + xt, \ \rm{when} \ t > s$$ Thus, the integral until $s$ converges, and the integral from $s$ oscillates faster than $t^2$, showing convergence.