Try to use integration by parts to evaluate the integral

If we let

Then we have

If we try to integrate by parts again with and then we’ll end up with another integral, this time with . That will continue, so we’ll just keep getting integrals that we can’t evaluate.

On the other hand, if we try letting

Then we have

Continuing along that route, we’ll keep getting integrals of where keeps getting larger.

In both cases, we keep getting increasing complicated integrals that we can never evaluate.

Perhaps the solution could be expersee

\int\frac{e^{x}}{x} = \frac{e^{x}}{x} + \int \frac{e^{x}}{x^2} = \frac{e^{x}}{x} + \frac{e^{x}}{x^2} + 2 \int \frac{e^{x}}{x^3} = \dots = e^x\sum\limits_{k=1}^{\infty} (k-1)! x^{-k}

(1)