How can one figure out whether $\int_1^{\infty} \frac{1+x+\ln x}{1+e^x} dx$ converges or diverges?

Question

How can one figure out whether $$\int_1^{\infty} \frac{1+x+\ln x}{1+e^x} dx$$ converges or diverges without evaluating the integral?

Answer 1

Notice that

$$\left|\frac{1+x+\ln x}{1+e^x}\right| = \frac{1+x+\ln x}{1+e^x} \le \frac{1+2x}{1+e^x} \le \frac{3x}{e^x}$$

for sufficiently large $x$.

---

It is easy to show that

$$\int_1^\infty \frac{3x}{e^x} dx$$ converges.