I want to investigate the convergence behavior of $$\int_{0}^{\infty} \cos(x^r)\, dx \hspace{5mm} \textrm{and} \hspace{5mm} \int_{\pi}^{\infty} \left(\frac{\cos x}{\log x}\right)\arctan\lfloor x\rfloor dx.$$ My theoretical tools are Abel's test and Dirichlet's test: Say I have an integral of the form $$\int_{a}^{b}f\cdot g \hspace{1.5mm} dx$$ with improperness (vertical or horizontal asymptote) at $b$.
Abel's test guarantees convergence if $g$ is monotone and bounded on $(a,b)$, and $\int_{a}^{b}f $ converges. Dirichlet's test guarantees convergence if $g$ is monotone on $(a,b)$ and $\displaystyle\lim_{x\to b} g(x) = 0$, and $\displaystyle\lim_{\beta \to b}$ $\int_{a}^{\beta}f $ is bounded.
For the first integral $\displaystyle\int_{0}^{\infty} \cos(x^r)\, dx $ I'm guessing a substitution $t = x^r $ will give me an expression of the form $f\cdot g$ with $\cos(t)$ as my $f$. For the second integral $\displaystyle\int_{\pi}^{\infty} \dfrac{\cos x}{\log x}\arctan\lfloor x\rfloor\, dx$, I'm (even more) clueless. Help please?
Hint for the second problem: A variant of Abel/Dirichlet for sums is the following: If $a_n$ is a monotonic convergent sequence and $\sum b_n$ converges, then $\sum a_nb_n$ converges. Proof sketch: Use the Cauchy criterion for the partial sums, sum by parts, ...
In your problem, think of $a_n = \arctan n,$ $b_n = \int_n^{n+1}\cos x/(\ln x)\, dx,$ $n=2,3,\dots $