TITLE: Asymptotic expansion of the integral $\int_2^x \frac{e^t}{t} dt$ for $x \to \infty$
QUESTION [5 upvotes]: Hello I wonder if there is any asymptotics known for such integral:
$$
   I(x) = \int_2^x \frac{e^t}{t} dt \qquad\text{when $ x\to+\infty $}.
$$
Thank you very much.

REPLY [8 votes]: Apply partial integration:
$$I(x) = \int_2^x \frac{e^t}{t} dt = \frac{e^x}{x}-\frac{e^2}{2}+\int_2^x \frac{e^t}{t^2}dt=\frac{e^x}{x}+\frac{e^x}{x^2}-\frac{3e^2}{4}-2\int_2^x \frac{e^t}{t^3}dt,$$ so $$\frac{e^x}{x}-\frac{e^2}{2}<I(x) < \frac{e^x}{x}+\frac{e^x}{x^2}-\frac{3e^2}{4}$$ and $$\lim\limits_{x \rightarrow \infty} I(x) / (e^x/x)=1.$$