$\int x e^{x^{2}} d x$

Step 1 ：Given the integral \(\int x e^{x^{2}} dx\)

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

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

Step 4 ：\(x = x\)

Step 5 ：\(f = x*exp(x**2)\)

Step 6 ：\(u = x**2\)

Step 7 ：\(du = 2*x\)

Step 8 ：\(f_u = exp(x**2)/2\)

Step 9 ：\(integral = \sqrt{\pi}*erfi(x)/4\)

Step 10 ：The integral of the function \(x e^{x^{2}}\) with respect to \(x\) is \(\frac{\sqrt{\pi} erfi(x)}{4} + C\), where \(erfi(x)\) is the imaginary error function and \(C\) is the constant of integration

Step 11 ：Final Answer: \(\boxed{\frac{\sqrt{\pi} erfi(x)}{4} + C}\)