Answer
$\dfrac{5 \sqrt 5+9}{6}$
Work Step by Step
Consider the integral such as: $\int_C f(x,y,z) ds=\int_{C_1} f(x,y,z) ds+\int_{C_2} f(x,y,z) ds$
and $\int_C f(x,y,z) ds=\int_{0}^{1} (2t)\sqrt{1^2+(2t)^2+(0)^2} dt+\int_{0}^{1} (2-t^2)\sqrt{0^2+0^2+1^2} dt$
$\int_C f(x,y,z) ds=\int_{0}^{1} (2t) \sqrt {4t^2+1} dt+\int_{0}^{1} (2-t^2)dt=(\dfrac{1}{6})[(4t^2+1)]_{1}^{5}+[2t-\dfrac{t^3}{3}]_{0}^{1}$
Thus, $\int_C f(x,y,z) ds=\dfrac{(5 \sqrt 5-1)}{6}+\dfrac{5}{3}=\dfrac{5 \sqrt 5+9}{6}$
