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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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