# Thread: Integral of 5^x

1. ## Integral of 5^x

As title states, I cannot recall how to integrate 5^x. I've tried looking in the book but can't seem to locate any example of it. I know it has to do with "e" and "ln" but can't seem to remember exactly.

Steps would be great.

2. Originally Posted by Beeorz
As title states, I cannot recall how to integrate 5^x. I've tried looking in the book but can't seem to locate any example of it. I know it has to do with "e" and "ln" but can't seem to remember exactly.

Steps would be great.
Hint:
$\displaystyle \frac{d}{dx}5^x = ln(5) \cdot 5^x$

-Dan

3. Originally Posted by topsquark
Hint:
$\displaystyle \frac{d}{dx}5^x = ln(5) \cdot 5^x$

-Dan
Haha well that makes it easy. Was really looking for some type of general rule so that I will be able to do just about any and all.

So the rule for any variable raised to a constant is to multiple it by ln(constant)??

4. Originally Posted by Beeorz
As title states, I cannot recall how to integrate 5^x. I've tried looking in the book but can't seem to locate any example of it. I know it has to do with "e" and "ln" but can't seem to remember exactly.

Steps would be great.
Let $\displaystyle 5^x=y$

taking natural log, $\displaystyle x \ln5 = \ln y$

so, $\displaystyle \ln 5.dx = \frac{y}{1} dy$

$\displaystyle \int 5^x ~dx = \frac{1}{\ln 5} \int \frac{y}{y}~dy$

$\displaystyle = \frac{1}{\ln 5} \int ~dy$

$\displaystyle = \frac{1}{\ln 5} y +c$

$\displaystyle = \frac{1}{\ln 5}(5^x) +c$

$\displaystyle = \frac{5^x}{\ln 5}+c$

in general, $\displaystyle \boxed{\int a^x ~dx =\frac{a^x}{\ln a} +c}, \text {where a is any constant}$