What is the integral of ##e^(7x)##?

It’s ##1/7e^(7x)##
What you want to calculate is:
##int e^(7x)dx##
We’re going to use .
Let ##u = 7x##
Differentiate (derivative) both parts:
##du = 7dx##
##(du)/7 = dx##
Now we can replace everything in the integral:
##int 1/7 e^u du##
Bring the constant upfront
##1/7 int e^u du##
The integral of ##e^u## is simply ##e^u##
##1/7e^u##
And replace the ##u## back
##1/7e^(7x)##
There’s also a shortcut you can use:
Whenever you have a function of which you know the integral ##f(x)##, but it has a different argument
##=>## the function is in the form ##f(ax+b)##
If you want to integrate this, it is always equal to ##1/a*F(ax+b)##, where ##F## is the integral of the regular ##f(x)## function.
In this case:
##f(x) = e^x##
##F(x) = int e^x dx = e^x##
##a = 7##
##b = 0##
##f(ax+b) = e^(7x)##
=> ## int e^(7x)dx = 1/a*F(ax+b) = 1/7*e^(7x) ##
If you use it more often, you will be able to do all these steps in your head.
Good luck!