Question

How to integrate $x{e}^x$?

Answer 1

Integrate the given expression :

Let $\mathrm{I}=\int x{e}^{x}dx$

We will use the method of by parts to solve the integration.

We now that

$\int u\left(x\right)v\left(x\right)dv=u\left(x\right)\int v\left(x\right)dx-\int \left[\frac{du\left(x\right)}{dx}\int v\left(x\right)dx\right]dx$

Let ,

1. $u\left(x\right)=x$
2. $v\left(x\right)=e^x$

Therefore,

$$\begin{gathered} \mathrm{I}=\int x{e}^{x}dx \\ =x\int e^{x}dx-\int \left[\frac{dx}{dx}\int e^{x}dx\right]dx \\ =x{e}^x-\int \frac{dx}{dx}·e^{x}dx\left[\because \int e^x=e^x+c\right] \\ =x{e}^x-\int e^{x}dx\left[\because \frac{dx}{dx}=1\right] \\ =x{e}^x-e^x+c \end{gathered}$$

$\therefore \int x{e}^{x}dx=x{e}^x-e^x+c$

Hence, the value of the integrating $\int x{e}^{x}dx$ is $x{e}^x-e^x+c$