Integrate 5^x
![Integrate 5^x](integrate-5_x/step-1.gif)
To integrate 5^x, also written as ∫ 5x dx, we first recognise that this is an exponential and will require the base change technique.
![Changing the base technique.](integrate-5_x/step-2.gif)
At this step, we change the base as shown above.
![Taking logs on both sides.](integrate-5_x/step-3.gif)
We take log to the base e on both sides.
![simplifying](integrate-5_x/step-4.gif)
At this step, we find that lne=1, which simplifies the above expression.
![Expression for z](integrate-5_x/step-5.gif)
We now have an expression for z.
![substitution of z.](integrate-5_x/step-6.gif)
We substitute for z in the expression 5^x=e^z to give the above expression.
![Integrating the easier version.](integrate-5_x/step-7.gif)
As you can see, we now have a simpler integration, which is the same thing.
![answer.](integrate-5_x/step-8.gif)
We integrate the simpler version, where C is the integration constant.
![Using substitution to simplify.](integrate-5_x/step-9.gif)
Since e^xln5=5^x, which we found earlier, we can substitute that to give the above expression.
![Final Answer - Top Marks!](integrate-5_x/step-10.gif)
Finally, this is the result.