integral of constant raised to a power

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

$\displaystyle \int{4^x dx}$

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.

1 Attachment(s)

Re: integral of constant raised to a power

Re: integral of constant raised to a power

Quote:

Originally Posted by

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

$\displaystyle \int{4^x dx}$

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

Re: integral of constant raised to a power

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

- Hollywood

Re: integral of constant raised to a power

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

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

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.

Re: integral of constant raised to a power

Yes,

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

is correct.

- Hollywood