<s>[INST] How can I study the convergence of the improper integral $\int_{2}^{\infty} \frac{\arctan(x+1)+x}{2^x+3^x}\, \mathrm dx$? I need to study the convergence of the following improper integral:
$$\int_{2}^{\infty} \dfrac{\arctan(x+1)+x}{2^x+3^x}\, \mathrm dx$$
I did the following:
$$ -\dfrac{\pi}{2} < \arctan(x+1)  < \dfrac{\pi}{2}   \\
\implies -\dfrac{\pi}{2} + x < \arctan(x+1) +x < \dfrac{\pi}{2} +x   \\
\implies   \dfrac{-\dfrac{\pi}{2} + x}{2^x+3^x} <  \dfrac{\arctan(x+1) +x}{2^x+3^x}  < \dfrac{\dfrac{\pi}{2} + x}{2^x+3^x}   \\
$$
I planned to integrate the inequality and then using the comparison criterion to proof its convergence. However, the idea did not work for me.
 [/INST] The function $x\mapsto \lvert\arctan(x+1)\rvert$ is globally bounded by $\frac\pi2$ so your integral is bounded in absolute value by $$\int_0^\infty \frac{\frac\pi2+x}{2^x+3^x}\,\mathrm dx<\infty.$$
Note that the last integral is finite because $$\int_0^\infty \frac{P(x)}{b^x}\,\mathrm dx$$ is finite for all polynomials $P$ and numbers $b>1$. This can be proven for example by noting that $$b^x\geq 1+C x^{\mathrm{deg}(P)},$$ where $C=\frac{\ln(b)^{n+2}}{(n+2)!}$ and $\mathrm {deg}(P)$ is the degree of the polynomial $P$.
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