$\int \frac{5}{6+5 x} d x, x \neq-\frac{6}{5}$

Step 1 ：Given the integral problem: \(\int \frac{5}{6+5 x} d x\), where \(x \neq -\frac{6}{5}\)

Step 2 ：We recognize this as a simple integral problem. The integral of a function of the form \(\frac{a}{b+cx}\) is \(\frac{a}{c} \ln|b+cx|\).

Step 3 ：Applying this formula to our problem, we get the integral of the function \(\frac{5}{6+5x}\) as \(\ln|6+5x|\).

Step 4 ：However, we need to divide this by the coefficient of \(x\) in the denominator, which is 5.

Step 5 ：So, the final answer is \(\frac{1}{5}\ln|6+5x|\).

Step 6 ：Final Answer: \(\boxed{\frac{1}{5}\ln|6+5x|}\)