Question

Solve the integral:$\int \frac{x}{e^x}dx$

Short answer

The required answer is$-x{e}^x-e^x+c$.

Step-by-step solution

 Step 1. Given information 

We have given integral is $\int \frac{x}{e^x}dx$.

Step 2. Solve the integration by parts .  

We have, $u=x,dv=\frac{dx}{e^x}$

$$\begin{gathered} u=x \\ du=dx \end{gathered}$$

and

$$\begin{gathered} dv=\frac{dx}{e^x} \\ v=\int \frac{dx}{e^x} \\ v=-\frac{1}{e^x} \end{gathered}$$

The formula of integration by parts is $\int udv=uv-\int vdu$

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

Step 3. Integration by parts. 

We have, $$\begin{gathered} u=-x \\ du=-dx \end{gathered}$$

So,

$$\begin{gathered} \left(-\frac{x}{e^x}\right)+\int e^{-x}du \\ =\left(-\frac{x}{e^x}\right)-e^{-x}+c \\ =-x{e}^x-e^x+c \end{gathered}$$