$\int \frac{2 x+1}{x^{2}-x-2} d x$

Step 1 ：First, let's find the partial fraction decomposition of the given function: $\frac{2x+1}{x^2-x-2} = \frac{1}{3(x+1)} + \frac{5}{3(x-2)}$

Step 2 ：Now that we have the partial fraction decomposition, we can integrate each term separately: $\int \frac{1}{3(x+1)} dx + \int \frac{5}{3(x-2)} dx$

Step 3 ：Integrate the first term: $\int \frac{1}{3(x+1)} dx = \frac{1}{3} \ln |3x+3| + C_1$

Step 4 ：Integrate the second term: $\int \frac{5}{3(x-2)} dx = \frac{5}{3} \ln |3x-6| + C_2$

Step 5 ：Combine the two integrals: $\frac{1}{3} \ln |3x+3| + \frac{5}{3} \ln |3x-6| + C$

Step 6 ：\(\boxed{\frac{5}{3} \ln |3x - 6| + \frac{1}{3} \ln |3x + 3| + C}\)