<s>[INST] I need help to evaluate this integral with a constant $c$ Find the value of the constant $c$ for which the integral converges, and evaluate the integral:
$$\int_0^\infty \left(\frac{9x}{x^2+1}-\frac{9c}{2x+1}\right)dx$$
 [/INST] We have
\begin{align}
\int_0^a \left(\dfrac{9x}{x^2+1} - \dfrac{9c}{2x+1}\right) dx & = \left. \left(\dfrac92 \log\left(x^2+1 \right) - \dfrac{9c}2 \log(2x+1) \right) \right \vert_0^a\\
& = \dfrac{9}2 \log \left(\dfrac{a^2+1}{(2a+1)^c}\right)
\end{align}
Given this, as $a \to \infty$, what happens if $c<2$ or $c>2$? Once you get the $c$ from this, use this $c$ and let $a \to \infty$, to get the answer.
</s>