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