= Solution
On the first segment, parameterize $\mathbf r_1(t)=(t,t,0)^T$, $0\leq t\leq1$. The <dot products> with $\mathbf r_1'=(1,1,0)^T$ are
$$
\mathbf F(\mathbf r_1)\cdot\mathbf r_1'=2t,\qquad
\mathbf G(\mathbf r_1)\cdot\mathbf r_1'=2t\cos(t^2).
$$
Their <line integrals> are one and $\sin1$, respectively.
On the second segment take $\mathbf r_2(t)=(1,1,t)^T$, with <tangent vector> $(0,0,1)^T$. The corresponding <integrands> are $a-2+2t$ and $c$, whose <integrals> are $a-1$ and $c$. Adding the segments yields
$$
\boxed{\int_{C_1}\mathbf F\cdot d\mathbf r=a,\qquad
\int_{C_1}\mathbf G\cdot d\mathbf r=\sin1+c}.
$$
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