$\int x e^{5 x} d x$

Step 1 ：Given the integral \(\int x e^{5 x} d x\)

Step 2 ：We can use the method of integration by parts, which is given by the formula \(\int u dv = uv - \int v du\)

Step 3 ：Let's choose \(u = x\) and \(dv = e^{5x} dx\)

Step 4 ：Then we find \(du = 1\) and \(v = \frac{e^{5x}}{5}\)

Step 5 ：Substitute these into the formula, we get \(uv - \int v du = x \cdot \frac{e^{5x}}{5} - \int \frac{e^{5x}}{5} \cdot 1 dx\)

Step 6 ：Simplify the integral on the right, we get \(x \cdot \frac{e^{5x}}{5} - \frac{e^{5x}}{25}\)

Step 7 ：Finally, don't forget to add the constant of integration, so the final answer is \(\boxed{\frac{1}{5}x e^{5x} - \frac{1}{25} e^{5x} + C}\)