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I had this show up in a problem and I went completely blank. How do I compute

$$\int a^t \mathrm{d}t$$

where $a$ is some constant?

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$a^t = e^{t \ln a}$ – mrf Nov 17 '12 at 22:06
Thanks for the responses but I found the answer here: – Korgan Rivera Nov 17 '12 at 22:07
Why anyone would bother to downvote this. -_- – Korgan Rivera Nov 18 '12 at 3:10
Perhaps the downvote came from the fact that this is a sufficiently simple integral that it is (as you found) easy to find in almost any table of integrals. – Rick Decker Nov 18 '12 at 17:07
@RickDecker Everything's easy when you know how. I didn't have access to a table of integrals at the time and, even if I did, I wanted to learn how to work it out. What a ridiculous reason to downvote. – Korgan Rivera May 23 '13 at 16:49
up vote 4 down vote accepted

$a^t = e^{t \ln a}$.

From there, $\int a^t dt = \int e^{t \ln a} dt$, which is trivial to integrate. The answer is $$\frac{a^t}{\ln a} + k$$

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Note that $a^t = (e^{\log(a)})^t = e^{t \log (a)}$ and $$\int e^{bt} dt = \dfrac{e^{bt}}{b} + \text{ constant}$$

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Since $\dfrac{\text{d}}{\text{dt}} a^t=a^t\ln(a),$ we have $$\int a^t \text{dt}=\dfrac{a^t}{\ln(a)}+C$$

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$$\int a^tdt=1/\log(a) \cdot \int a^t \log(a)dt=a^t/ \log(a)+C$$ This is because the derivative of $a^t$ is $a^t \log(a)$ .

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Try \log(a) instead of log(a). – leo Nov 17 '12 at 22:25

Use the fact that $$\frac{\text{d}(a^t)}{\text{d}t}=a^t\ln(a)$$

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