Exponential integration (1 Viewer)

[Finx]

∫(3^x) dx

It's probably some simple rule I missed out on. Thanks in advance!

 

[lyounamu]

∫(3^x) dx

It's probably some simple rule I missed out on. Thanks in advance!

d/dx( 3^x) = ln3 3^x

so int 3^x dx = 1/ln3 int ln3 3^x dx = 1/ln3 3^x + c

 

[Trebla]

The rule d(3^x)/dx = 3^x ln 3 comes from the following process:
Let y = 3^x
ln y = x ln 3
x = ln y / ln 3
dx/dy = 1 / y ln 3
dy/dx = y ln 3
= 3^x ln 3
.: d(3^x)/dx = 3^x ln 3

So:
∫ 3^x ln 3 dx = 3^x + c₁
∫ 3^x dx = 3^x / ln 3 + c₂