Question

Find the value of $\int x{e}^{x}dx$

Answer 1

Finding the value of $\int x{e}^{x}dx$

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

We know that,

$\int uvdx=u\int vdx-\int \left(\frac{d}{dx}\right(u)\int vdx)dx$

So,

$$\begin{gathered} I=x\int e^{x}dx-\int \frac{d\left(x\right)}{dx}(\int e^{x}dx)dx \\ =x{e}^x-\int 1.e^{x}dx \\ =x{e}^x-e^x+c \end{gathered}$$

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