The integral 2𝜋 4xe−x2 dx can be done with the substitution u = and du = dx.

The integral ∫2π4xe^(-x^2) dx can be solved using the method of substitution.

Step 1: Choose a substitution. In this case, we let u = x^2.

Step 2: Differentiate u with respect to x to get du/dx. So, du/dx = 2x.

Step 3: Rearrange the equation to solve for dx. So, dx = du/(2x).

Step 4: Substitute u and dx into the integral. The integral becomes ∫2π4x e^(-u) du/(2x).

Step 5: Simplify the integral. The x's cancel out and the 2's combine to give ∫2π2e^(-u) du.

Step 6: Now, you can integrate. The integral of e^(-u) with respect to u is -e^(-u). So, the integral becomes -2π2e^(-u) + C, where C is the constant of integration.

Step 7: Substitute x^2 back in for u to get the final answer: -2π2e^(-x^2) + C.