Question over improper integral.

Question

I need to check whether $\int_0^\infty \frac{2^t+t}{3^t+t}dt$ this integral is proper or improper. I know that this is an improper integral of first kind. But if I take $I=\int_0^\infty \frac{2^t+t}{3^t+t}dt=\lim_{x\to \infty}\int_0^x \frac{2^t+t}{3^t+t}dt$ then how to integrate this.

Answer 1

For very large values of $t$, there is no problem since $$\frac{2^t+t}{3^t+t}\sim \left(\frac{2}{3}\right)^t$$ Close to $t=0$, using Taylor $$\frac{2^t+t}{3^t+t}=1+\log\left(\frac{2}{3}\right)t+O(t^2)$$ No problem either.