Rewriting Riemann sum as integral

Question

Can anybody tell me how to rewrite this sum limit as integral I am struggling with converting this equation into definite integral form

$$\lim_{n \to \infty} \sum_{k=1}^n \frac{\ln{\left(5k+n\right)}-\ln{(n)}}{n}$$

Here is the original question: enter image description here

Answer 1

Rewrite the summation using log properties: $$\lim_{n \to \infty} \sum_{k=1}^n \frac{\ln{\left(\frac{5k}{n}+1\right)}}{n}$$ This way, $x=\frac{k}{n}$ and $dx=\frac{1}{n}$. $$\int_0^1 \ln{\left(5x+1\right)} \; dx$$ Try finishing it.