$=\int_0^1 \dfrac{\sqrt{x+3}-\sqrt{x+1}}{2}dx=\dfrac{1}{2} \int_0^1 \sqrt{x+3} d(x+3) -\dfrac{1}{2}\int_0^1 \sqrt{x+1} d(x+1)$$=\dfrac{1}{3} \bigg (\sqrt{(x+3)^3}-\sqrt{(x+1)^3} \bigg ) \bigg |_0^1 = 3-\sqrt 3 -\dfrac{2\sqrt 2}{3}$
$=\int_0^1 \dfrac{\sqrt{x+3}-\sqrt{x+1}}{2}dx=\dfrac{1}{2} \int_0^1 \sqrt{x+3} d(x+3) -\dfrac{1}{2}\int_0^1 \sqrt{x+1} d(x+1)$$=\sqrt{(x+3)^3}-\sqrt{(x+1)^3}+C$
$=\int_0^1 \dfrac{\sqrt{x+3}-\sqrt{x+1}}{2}dx=\dfrac{1}{2} \int_0^1 \sqrt{x+3} d(x+3) -\dfrac{1}{2}\int_0^1 \sqrt{x+1} d(x+1)$$=\
dfrac{1}{3} \bigg (\sqrt{(x+3)^3}-\sqrt{(x+1)^3}
\bigg ) \bigg |_0^1 = 3-\sqrt 3 -\dfrac{2\sqrt 2}{3}$