Aloha :)
$$I_a=\int\limits_C2xyz^2\,ds=\!\int\limits_0^12x(t)y(t)z^2(t)\left\|\frac{d\vec s(t)}{dt}\right\|\,dt=\int\limits_0^12\,t^2\,2t\,\frac{t^6}{9}\,\left\|\frac{d}{dt}\begin{pmatrix}t^2\\2t\\\frac{1}{3}t^3\end{pmatrix}\right\|\,dt$$$$\phantom{I_a}=\int\limits_0^1\frac{4}{9}t^9\left\|\begin{pmatrix}2t\\2\\t^2\end{pmatrix}\right\|\,dt=\int\limits_0^1\frac{4}{9}t^9\sqrt{4t^2+4+t^4}\,dt=\int\limits_0^1\frac{4}{9}t^9\sqrt{(t^2+2)^2}\,dt$$$$\phantom{I_a}=\int\limits_0^1\frac{4}{9}t^9(t^2+2)\,dt=\int\limits_0^1\left(\frac{4}{9}t^{11}+\frac{8}{9}t^9\right)\,dt=\left[\frac{1}{27}t^{12}+\frac{4}{45}t^{10}\right]_0^1=\frac{17}{135}$$
$$I_b=\int\limits_C\frac{x}{1+x^2+y^2}\,ds=\int\limits_0^\pi\frac{x(t)}{1+x^2(t)+y^2(t)}\left\|\frac{d\vec s(t)}{dt}\right\|\,dt$$$$\phantom{I_b}=\int\limits_0^\pi\frac{t\cos t}{\sqrt{1+t^2\cos^2t+t^2\sin^2t}}\left\|\frac{d}{dt}\begin{pmatrix}t\cos t\\t\sin t\end{pmatrix}\right\|\,dt$$$$\phantom{I_b}=\int\limits_0^\pi\frac{t\cos t}{\sqrt{1+t^2}}\left\|\begin{pmatrix}\cos t-t\sin t\\\sin t+t\cos t\end{pmatrix}\right\|\,dt=\int\limits_0^\pi\frac{t\cos t}{\sqrt{1+t^2}}\sqrt{1+t^2}\,dt$$$$\phantom{I_b}=\int\limits_0^\pi \underbrace{t}_{=u}\,\underbrace{\cos t}_{=v'}\,dt=\left[\underbrace{t}_{=u}\,\underbrace{\sin t}_{=v}\right]_0^\pi-\int\limits_0^\pi \underbrace{1}_{=u'}\,\underbrace{\sin t}_{=v}\,dt=0+\left[\cos t\right]_0^\pi=-2$$
