Evaluate the definite integral ${\int }_1^{25}\frac{5}{\sqrt{x}}dx$ using the Fundamental Theorem of Calculus, Part 2.

Compute ${\int }_4^{16}\frac{x-8}{\sqrt{x}}dx$ using the Fundamental Theorem of Calculus, Part 2.

Evaluate $\frac{d}{dt}{\int }_{\tan t}^3\frac{dx}{x^5+3}$.

Evaluate ${\lim }_{x\to 3}\frac{{\int }_3^x\sqrt{t^2-2t+6}dt}{x^2-9}$.

Let $g\left(x\right)={\int }_2^x{t}^2(t-1)(t-5)dt$. Find the $x$-coordinates of the critical points of $g\left(x\right)$ and determine the intervals where $g$ is increasing or decreasing.