$ f(x)=\left ( x-\sqrt{x}+1 \right )\left (\sqrt{x}+1 \right )=\left ( \sqrt{x} \right )^{3}+1=x^{\frac{3}{2}}+1$
$\Rightarrow\int
f(x)dx=\int(x^{\frac{3}{2}}+1)dx=\frac{2}{5}x^{\frac{5}{2}}+x+C$
$g(x)=\cot x+5\left ( \sqrt{2x+1}-\sqrt{2x-1} \right )=\frac{\left ( \sin x \right )^{'} }{\sin x}+5\left ( 2x+1 \right )^{\frac{1}{2}}-5\left ( 2x-1 \right )^{\frac{1}{2}}$
$\Rightarrow\int g(x)dx=\ln|\sin
x|+\frac{5}{3}(2x+1)^{\frac{3}{2}}-\frac{5}{3}(2x-1)^{\frac{3}{2}}+C$