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"The hands move, the lips move—Ideas gush from his words, And his eyes devour! He is an island of Selfdom."

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:

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

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