<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] Just observe that RHS is less than $$\frac{\frac{\pi}{2}+x}{3^x}$$Since $$\int_2^\infty\frac{1}{3^x} \,dx$$clearly converges, it is enough to show that$$\int_2^\infty\frac{x}{3^x} \,dx$$ converges which can be done using limit comparison test (with $g(x)=\frac{1}{e^x}$).
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