Evaluate the integral using integration by parts.
\[
\int x e^{5 x} d x
\]

Step 1 ：Let's solve the integral \(\int x e^{5 x} dx\) using the method of integration by parts. The formula for integration by parts is \(\int u dv = uv - \int v du\).

Step 2 ：We choose \(u = x\) and \(dv = e^{5x} dx\).

Step 3 ：We then find \(du\) and \(v\). Here, \(du = 1\) and \(v = \frac{e^{5x}}{5}\).

Step 4 ：We calculate \(uv = \frac{x e^{5x}}{5}\) and \(\int v du = \frac{e^{5x}}{25}\).

Step 5 ：Substitute these values into the integration by parts formula, we get \(\int x e^{5x} dx = \frac{x e^{5x}}{5} - \frac{e^{5x}}{25}\).

Step 6 ：Finally, we add the constant of integration \(C\) to get the final answer.

Step 7 ：\(\boxed{\frac{x e^{5x}}{5} - \frac{e^{5x}}{25} + C}\)