Evaluate a definite integral involving a square root using FTC Part 2

By the Fundamental Theorem of Calculus Part 2, since $F(x) = 2\sqrt{x}$ is an antiderivative of $f(x) = \frac{1}{\sqrt{x}}$, we have $\int_1^4 \frac{1}{\sqrt{x}}\,dx = F(x)\Big|_1^4 = F(4) - F(1)$. This reduces the problem to evaluating $F$ at the two endpoints.

Compute $F(4) = 2\sqrt{4} = 2(2) = 4$ and $F(1) = 2\sqrt{1} = 2(1) = 2$. Subtracting gives $F(4) - F(1) = 4 - 2 = 2$. Thus $\int_1^4 \frac{1}{\sqrt{x}}\,dx = 2$.