Solve $\int x{e}^{x}dx$

Step-by-step solution

1. Step 1 · Choose parts
   $$u=x,dv=e^{x}dx$$

   Differentiating x simplifies it; e^x integrates to itself.

2. Step 2 · Compute du and v
   $$du=dx,v=e^x$$

   Prepare both pieces of the parts formula.

3. Step 3 · Integration by parts
   $$x{e}^x-\int e^{x}dx$$

   ∫u dv = uv − ∫v du.

4. Step 4 · Finish the integral
   $$x{e}^x-e^x+C$$

   Differentiating the result returns xe^x, confirming the answer.