Question

How do you integrate by parts: `int x e^3xdx`?

Answer 1

This seems to be:

`int xe^(3x)dx`

`e^3(x)` wouldn't make sense (`e` is a constant, not a function), and `e^3x ne (ex)^3` (improper implications from `trig^n(x) = (trigx)^n`). `e^(x^3)` would be very advanced to integrate, and would not be remotely easy by integration by parts. `x^2e^3` would be way too simple.

Assuming so...

Let:
`u = x`
`du = 1dx`
`dv = e^(3x)dx`
`v = 1/3e^(3x)`

`= uv - intvdu`

`= x/3e^(3x) - 1/3inte^(3x)dx`

`= x/3e^(3x) - 1/3[1/3e^(3x)] + C]`

`= x/3e^(3x) - 1/9e^(3x) + C`

or

`= e^(3x)/9 (3x - 1) + C`