integral of constant raised to a power

**infraRed** · January 27th 2013, 06:19 PM

Hi. What is the integral of a constant raised to a variable power? For example:

$\int{4^x dx}$

I imagined first that perhaps you could treat it like a variable raised to a constant power, but I'm not sure if that is correct, and my results didn't seem quite right when I looked at a graph of the function and an antiderivative together.

**ibdutt** · January 27th 2013, 06:37 PM

**Plato** · January 27th 2013, 06:38 PM

Originally Posted by infraRed

Hi. What is the integral of a constant raised to a variable power? For example:

$\int{4^x dx}$

If $y=4^x$ then $y'=4^x(log(4))$ so what is the answer to your question?

**hollywood** · January 27th 2013, 08:19 PM

It might be even easier to recognize that $4^x = e^{x\ln{4}}$ , which is e to a constant times x. I'm guessing you know how to integrate that....

- Hollywood

**infraRed** · January 27th 2013, 08:36 PM

I found in my calculus textbook a table of basic integration rules, which states that

$\int{a^u du} = (\frac{1}{\ln{a}})a^u + C$

Since I'm short on time, it's easiest just to go with that. I imagine that the hints provided above were leading me in that direction.

**hollywood** · January 27th 2013, 08:47 PM

Yes,

$\int{4^x dx} = \int{e^{(\ln{4})x} dx} = \frac{1}{\ln{4}} e^{(\ln{4})x} + C = \frac{1}{\ln{4}} 4^x + C$

is correct.

- Hollywood

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