How would I integrate $e^{e^x}$?

Question

Is there a way to integrate:

$e^{e^x}$

without using a Taylor or McLaurin Series expansion?

Answer 1

$$\int e^{e^x}\,\mathrm{d}x$$

Let $t=e^x\implies\mathrm{d}t=e^x\mathrm{d}x=t\mathrm{d}x\implies \mathrm{d}x=\frac{\mathrm{d}t}{t}$

$$\int \frac{e^t\mathrm{d}t}{t}=\int\frac{\mathrm{d}(e^t)}{t}$$

$$\operatorname{Ei}(t)+c=\operatorname{Ei}(e^x)+c$$

$\operatorname{Ei}(x)$ is the exponential integral.