Answer
$$\int \frac{x+2 \sqrt{x-1}}{2 x \sqrt{x-1}} d x=\sqrt{(x-1)}+\ln x+c $$
Work Step by Step
Given $$\int \frac{x+2 \sqrt{x-1}}{2 x \sqrt{x-1}} d x $$
So, we have
\begin{aligned}
I&=\int \frac{x+2 \sqrt{x-1}}{2 x \sqrt{x-1}} d x\\
&=\int \frac{x d x}{2 x \sqrt{x-1}}+\int \frac{2 \sqrt{x-1} d x}{2 x \sqrt{x-1}} d x\\
&=\int \frac{d x}{2 \sqrt{x-1}}+\int \frac{d x}{x}\\
&=\frac{(x-1)^{-1 / 2+1}}{2(1/2)}+\ln x+c\\
&=\sqrt{(x-1)}+\ln x+c
\end{aligned}
