1. **Stating the problem:** We want to evaluate or understand the integral expression for the vector field $ \vec{E} $ given by $$\vec{E} = \frac{1}{4 \pi \epsilon_0} \int_{-L/2}^{1} \frac{1}{(r^2 + z^2)^{3/2}} \, dt$$ 2. **Understanding the integral:** This integral represents a component of the electric field $ \vec{E} $ due to a distribution along the variable $ t $ from $ -L/2 $ to $ 1 $. 3. **Formula and rules:** The constant $ \frac{1}{4 \pi \epsilon_0} $ is Coulomb's constant factor in electrostatics. The integrand $ \frac{1}{(r^2 + z^2)^{3/2}} $ suggests a dependence on the distance from the source point to the field point, where $ r $ and $ z $ are spatial coordinates or functions of $ t $. 4. **Next steps:** To evaluate the integral, we need explicit expressions for $ r $ and $ z $ as functions of $ t $. Without these, the integral cannot be simplified further. 5. **Summary:** The integral expression is set up correctly for calculating $ \vec{E} $ but requires more information about $ r(t) $ and $ z(t) $ to proceed with evaluation. **Final answer:** The integral expression for $ \vec{E} $ is $$\vec{E} = \frac{1}{4 \pi \epsilon_0} \int_{-L/2}^{1} \frac{1}{(r^2 + z^2)^{3/2}} \, dt$$ where $ r $ and $ z $ must be specified to evaluate further.