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"The stone cannot know why the chisel cleaves it; the iron cannot know why the fire scorches it. When thy life is cleft and scorched, when death and despair leap at thee, beat not thy breast and curse thy evil fate, but thank the Builder for the trials that shape thee."

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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