Answer
$2$
Work Step by Step
Step 1. The limit of the Riemann sum leads to a definite integral, and from the expression given in the exercise, we can identify that $f(x)=(2x-1)^{-1/2}$ and the interval is $[1,5]$.
Step 2. We can write the integral as
$\int_1^5f(x)dx=\int_1^5(2x-1)^{-1/2}dx$
Step 3. Using substitution, let $u=2x-1$, we have $du=2dx$ with $x\to1, u\to1$ and $x\to5, u\to9$
Step 4. We have
$\int_1^5f(x)dx=\frac{1}{2}\int_1^9(u)^{-1/2}du=u^{1/2}|_1^9=\sqrt 9-1=2$
