Here's the integral again: $$\int_{\vec{\gamma_j}}\langle\vec{v_k},d\vec{x}\rangle$$
Here's what I know about $\vec{\gamma_1},\vec{\gamma_2}:[0,1]\rightarrow\mathbb{R^3}$
$$\vec{\gamma_1}(t) = \left(\! \begin{array}{c} t \\ t+1 \\ t \end{array} \!\right) \;,\; \vec{\gamma_2}(t) = \left(\! \begin{array}{c} t \\ t^2+1 \\ t \end{array} \!\right) $$
And the vector fields $\vec{v_1},\vec{v_2},\vec{v_3}:\mathbb{R}\rightarrow\mathbb{R}$ are continuous and differentiable:
$$\vec{v_1}(x,y,z) = \left(\! \begin{array}{c} x^2-y \\ y^2+x \\ z \end{array} \!\right) \;,\;\vec{v_2}(x,y,z) = \left(\! \begin{array}{c} x^3-3xy^2 \\ y^3+2yx^2 \\ 5 \end{array} \!\right) \;,\;\vec{v_3}(x,y,z) = \left(\! \begin{array}{c} f'(x) \\ g'(y) \\ h'(z) \end{array} \!\right) $$
I'm new to the concept of calculating line integrals so any advice on how to move forward would be very appreciated.