How to compute $\int a^t \mathrm{d}t$?

Question

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?

Answer 1

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

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

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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