Question

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

Answer 1

It's `1/7e^(7x)`
What you want to calculate is:
`int e^(7x)dx`

We're going to use u-substitution .

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!