<s>[INST] Find values of constants a and b such that the given improper integral converges Find values of constants a and b such that:$$\int_{3}^\infty \left(\frac{ax+2}{x^2+3x}-\frac{b}{3x-2}\right) dx=k$$ then by partial fractions we get:
$$\lim_{N \to \infty} \int_{3}^N \left(\frac{2}{3}\frac{1}{x}+\frac{a-\frac{2}{3}}{x+3}-\frac{b}{3x-2}\right) dx=k$$
then
$$\lim_{N \to \infty} \left[\frac{2}{3}ln(x)+(a-\frac{2}{3})ln (x+3)-\frac{b}{3}ln(3x-2)\right]_3^N =k$$
By logarithm properties:
$$\lim_{N \to \infty} \left[ln\frac{(x)^\frac{2}{3}(x+3)^\left(a-\frac{2}{3}\right)}{(3x-2)\frac{b}{3}}\right]_3^N =k$$
Evaluating:
$$\lim_{N \to \infty} \left[ln\frac{(N)^\frac{2}{3}(N+3)^\left(a-\frac{2}{3}\right)}{(3N-2)^\frac{b}{3}}\right]-\left[ln\frac{(3)^\frac{2}{3}(6)^\left(a-\frac{2}{3}\right)}{(7)^\frac{b}{3}}\right] =k$$
But, I do not know what to do after this step. Any recommendation will be appreciated
 [/INST] Hint: It converges when 

$$ \lim _{ n\to \infty  } \ln { \left[ \sqrt [ 3 ]{ \frac { { n }^{ 2 }{ \left( 
n+3 \right)  }^{ 3a-2 } }{ { \left( 3n-2 \right)  }^{ b } }  }  \right]  } \\ \\ 2+3a-2=b\quad \Rightarrow b=3a $$

</s>