Evaluate. (Be sure to check by differentiating!) \[ \int \frac{1}{7+3 x} d x, x \neq-\frac{7}{3} \]

Step 1 ：The integral is in the form of \(\int \frac{1}{a+bx} dx\), which can be solved using the formula \(\frac{1}{b} \ln |a+bx| + C\). Here, \(a=7\) and \(b=3\). So, we can substitute these values into the formula to get the solution.

Step 2 ：Substituting the values we get, \(\frac{1}{3} \ln |3x + 7| + C\).

Step 3 ：Now, we will differentiate this result to check if it matches the original function.

Step 4 ：The derivative of the integral is the same as the original function, which confirms that the integral is correct.

Step 5 ：The integral of the function \(\frac{1}{7+3x}\) is \(\boxed{\frac{1}{3} \ln |3x + 7| + C}\).