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.
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.
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} and $\lim\limits_{x \rightarrow \infty} I(x) / (e^x/x)=1.$