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

Jun 9, 2015

This seems to be:

$\int x {e}^{3 x} \mathrm{dx}$

${e}^{3} \left(x\right)$ wouldn't make sense ($e$ is a constant, not a function), and ${e}^{3} x \ne {\left(e x\right)}^{3}$ (improper implications from $t r i {g}^{n} \left(x\right) = {\left(t r i g x\right)}^{n}$). ${e}^{{x}^{3}}$ would be very advanced to integrate, and would not be remotely easy by integration by parts. ${x}^{2} {e}^{3}$ would be way too simple.

Assuming so...

Let:
$u = x$
$\mathrm{du} = 1 \mathrm{dx}$
$\mathrm{dv} = {e}^{3 x} \mathrm{dx}$
$v = \frac{1}{3} {e}^{3 x}$

$= u v - \int v \mathrm{du}$

$= \frac{x}{3} {e}^{3 x} - \frac{1}{3} \int {e}^{3 x} \mathrm{dx}$

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

$= \frac{x}{3} {e}^{3 x} - \frac{1}{9} {e}^{3 x} + C$

or

$= {e}^{3 x} / 9 \left(3 x - 1\right) + C$