What is $ \int_0^{\infty} \left( e^{\frac{\ln x} {x}}-1 \right) dx$?

Question

As per what I have tried, I don't think that $ \int\left( e^{\frac{\ln x} {x}} \right) dx$ can be expressed in terms of elementary functions, but then I again I have just started learning integration by myself so i'm not sure.

Any help will be appreciated.

Answer 1

There is no need to find the indefinite integral (which can't be expressed in terms of standard mathematical functions).

Note that $e^t\geq 1+t$ for $t\in \mathbb{R}$ and therefore $$\int_e^{\infty} \left( e^{\frac{\ln x} {x}}-1 \right) dx\geq \int_e^{\infty} \frac{\ln x}{x}\, dx\geq \int_e^{\infty} \frac{1}{x}\, dx=+\infty.$$ Moreover $$\int_0^{e} \left( e^{\frac{\ln x} {x}}-1 \right) dx$$ is a finite number because the integrand function is continuous and bounded in $(0,1]$ (note that $\lim_{x\to 0^+}e^{\frac{\ln x} {x}}=0$.)

So what may we conclude?