# Archive — hermiene.net

"What did the tree learn from the earth to be able to talk with the sky?"

### May 16, 2004

I've been trying and trying to integrate $a^x$ but never made it. That is, until a class-mate of mine helped me. The answer was right in front of me, but I didn't see it. Anyway, here goes. We set it up:

$$\int{a^x} dx$$

Next, we transform it a little:

$$\int{e^{\ln a^x}} dx$$

The rest is pretty straight-forward:

$$\int{e^{x \ln a}} dx = \frac{1}{\ln a} e^{x \ln a} = \frac{1}{\ln a} e^{(\ln a^x)} = \frac{1}{\ln a} a^x = \frac{a^x}{\ln a}$$