Answer
$\dfrac{ 5 \sqrt 5+7\sqrt 2-1}{6}$
Work Step by Step
Here, we have
$\int_C f(x,y,z) ds=\int_{C_1} f(x,y,z) ds+\int_{C_2} f(x,y,z) ds$
$\int_C f(x,y,z) ds=\int_{0}^{1} 2t \sqrt {1+4t^2} dt+\int_{1}^{0}(t+\sqrt t) \sqrt 2 dt$
This implies that
$\dfrac{1}{4} [\dfrac{2(1+4t^2)^{3/2}}{3}]_0^1+\sqrt 2 [\dfrac{t^2}{2}+\dfrac{2t^{3/2}}{3}]_1^0=\dfrac{ 5 \sqrt 5+7\sqrt 2-1}{6}$
