how do you integrate 5^x? (1 Viewer)

[megaman64]

question in title

 

[m.jakaran]

You know how to integrate e^kx, so write 5^x in this form. i.e. e^(ln(5)x).

 

[Timske]

<a href="http://www.codecogs.com/eqnedit.php?latex=\dpi{100} \int 5^{x} dx = \int e^{\5lnx } dx = \frac{1}{\ln 5}e^{\5lnx }@plus;\textup{C} \\\\ \frac{1}{\ln 5}e^{\5lnx } @plus; \textup{C} = \frac{1}{\ln 5}5^{x} @plus; \textup{C}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\dpi{100} \int 5^{x} dx = \int e^{\5lnx } dx = \frac{1}{\ln 5}e^{\5lnx }+\textup{C} \\\\ \frac{1}{\ln 5}e^{\5lnx } + \textup{C} = \frac{1}{\ln 5}5^{x} + \textup{C}" title="\dpi{100} \int 5^{x} dx = \int e^{\5lnx } dx = \frac{1}{\ln 5}e^{\5lnx }+\textup{C} \\\\ \frac{1}{\ln 5}e^{\5lnx } + \textup{C} = \frac{1}{\ln 5}5^{x} + \textup{C}" /></a>

 

[Sy123]

 

[megaman64]

how would you differentiates it?

 

[Fus Ro Dah]

how would you differentiates it?

Differentiate both sides of what Sy123 has with respect to x, then make d/dx (5^x) the subject.

 

[Timske]

<a href="http://www.codecogs.com/eqnedit.php?latex=\dpi{100} \int e^{ax@plus;b} dx = \frac{1}{a}e^{ax@plus;b} @plus; C\\\\ \frac{d}{dx}~ e^{ax@plus;b} = ae^{ax@plus;b} \\\\ \therefore \frac{d}{dx}~ e^{ln5x} = ln5*e^{ln5x} = ln5*5^{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\dpi{100} \int e^{ax+b} dx = \frac{1}{a}e^{ax+b} + C\\\\ \frac{d}{dx}~ e^{ax+b} = ae^{ax+b} \\\\ \therefore \frac{d}{dx}~ e^{ln5x} = ln5*e^{ln5x} = ln5*5^{x}" title="\dpi{100} \int e^{ax+b} dx = \frac{1}{a}e^{ax+b} + C\\\\ \frac{d}{dx}~ e^{ax+b} = ae^{ax+b} \\\\ \therefore \frac{d}{dx}~ e^{ln5x} = ln5*e^{ln5x} = ln5*5^{x}" /></a>