Evaluate. Check by differentiating. \[ \int 7 x e^{6 x} d x \]

Step 1 ：Given the integral \(\int 7 x e^{6 x} d x\)

Step 2 ：We can solve this using integration by parts. The formula for integration by parts is \(\int udv = uv - \int vdu\)

Step 3 ：Let's choose \(u = x\) and \(dv = 7e^{6x} dx\)

Step 4 ：Then we find \(du\) and \(v\). \(du\) is the derivative of \(u\) with respect to \(x\), which is \(dx\). \(v\) is the integral of \(dv\), which is \(\frac{7}{6}e^{6x}\)

Step 5 ：Substitute \(u\), \(v\), \(du\), and \(dv\) into the integration by parts formula, we get \(\int 7 x e^{6 x} d x = \frac{7x e^{6x}}{6} - \int \frac{7e^{6x}}{6} dx\)

Step 6 ：Integrate \(\frac{7e^{6x}}{6} dx\) to get \(\frac{7e^{6x}}{36}\)

Step 7 ：Substitute this back into the equation, we get \(\int 7 x e^{6 x} d x = \frac{7x e^{6x}}{6} - \frac{7e^{6x}}{36}\)

Step 8 ：Finally, add the constant of integration \(C\) to get the final answer \(\int 7 x e^{6 x} d x = \frac{7x e^{6x}}{6} - \frac{7e^{6x}}{36} + C\)

Step 9 ：We can check this by differentiating the result and seeing if we get back the original integrand, \(7x e^{6x}\)

Step 10 ：\(\boxed{\frac{7x e^{6x}}{6} - \frac{7e^{6x}}{36} + C}\) is the final answer