Evaluate the integral using integration by parts.
\[
\int 3 x e^{2 x} d x
\]
\[
\int 3 x e^{2 x} d x=\square
\]

Step 1 ：Let's solve the integral \(\int 3 x e^{2 x} dx\) using the method of integration by parts. The formula for integration by parts is \(\int udv = uv - \int vdu\).

Step 2 ：We can choose \(u = 3x\) and \(dv = e^{2x} dx\).

Step 3 ：Then we need to find \(du\) and \(v\). \(du\) is the derivative of \(u\) and \(v\) is the integral of \(dv\).

Step 4 ：The derivative of \(u = 3x\) is \(du = 3 dx\).

Step 5 ：The integral of \(dv = e^{2x} dx\) is \(v = \frac{1}{2} e^{2x}\).

Step 6 ：Now we substitute \(u\), \(v\), \(du\), and \(dv\) into the formula \(\int udv = uv - \int vdu\).

Step 7 ：This gives us \(\int 3 x e^{2 x} dx = 3x \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} \cdot 3 dx\).

Step 8 ：Solving the remaining integral, we get \(\int \frac{1}{2} e^{2x} \cdot 3 dx = \frac{3}{4} e^{2x}\).

Step 9 ：Substituting this back into the equation, we get \(\int 3 x e^{2 x} dx = \frac{3}{2}x e^{2x} - \frac{3}{4} e^{2x}\).

Step 10 ：Finally, we add the constant of integration \(C\) to get the final answer.

Step 11 ：\(\boxed{\int 3 x e^{2 x} dx = \frac{3}{2}x e^{2x} - \frac{3}{4} e^{2x} + C}\)