Evaluate the improper integral $\int_{0}^{\infty} x e^{-x^{2}} d x$

Step 1 ：Given the improper integral \(\int_{0}^{\infty} x e^{-x^{2}} d x\)

Step 2 ：We can use the method of integration by substitution. Let's set \(u = x^2\), then \(du = 2x dx\)

Step 3 ：Rewrite the integral in terms of \(u\) and solve it

Step 4 ：The integral has been evaluated successfully. The result is \(\sqrt{\pi}/4\)

Step 5 ：Final Answer: \(\boxed{\frac{\sqrt{\pi}}{4}}\)