Integrating x(e^x): Step-by-Step Guide

[theRukus]

Homework Statement

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

The Attempt at a Solution

It's part of a larger question. I've got a midterm tomorrow and just realized I don't know the principles of integrating e^x other than that the integral of e^x is e^x. I've scoured my textbook and cannot find it. Could anyone provide insight?

 

[Char. Limit]

What you're looking for is integration by parts, which can be proven from the Product Rule. If

$$d(uv) = v du + u dv$$

Then subtract v du and integrate:

$$\int u dv = \int d(uv) - \int v du = uv - \int v du$$