\( E = \frac { 1 } { 2 \pi \cdot \varepsilon _ { 0 } } \int _ { \pi } ^ { \pi } \frac { x \cdot \frac { x \cdot d \theta } { \cos ^ { 2 } \theta \cdot \cos \theta } } { \frac { x } { \cos \theta } } \)

To evaluate the integral given in the expression for \( E \): \[ E = \frac{1}{2 \pi \cdot \varepsilon_0} \int_{\pi}^{\pi} \frac{x \cdot \frac{x \cdot d\theta}{\cos^2 \theta \cdot \cos \theta}}{\frac{x}{\cos \theta}} \] First, let's simplify the integrand. The expression inside the integral can be rewritten as follows: 1. The term \( \frac{x \cdot \frac{x \cdot d\theta}{\cos^2 \theta \cdot \cos \theta}}{\frac{x}{\cos \theta}} \) simplifies to: \[ \frac{x^2 \cdot d\theta}{\cos^2 \theta \cdot \cos \theta} \cdot \frac{\cos \theta}{x} = \frac{x \cdot d\theta}{\cos^2 \theta} \] Thus, the integral becomes: \[ E = \frac{1}{2 \pi \cdot \varepsilon_0} \int_{\pi}^{\pi} \frac{x \cdot d\theta}{\cos^2 \theta} \] However, notice that the limits of integration are from \( \pi \) to \( \pi \). This means that the integral evaluates to zero because the upper and lower limits are the same: \[ \int_{\pi}^{\pi} f(\theta) \, d\theta = 0 \] Therefore, we conclude that: \[ E = \frac{1}{2 \pi \cdot \varepsilon_0} \cdot 0 = 0 \] Thus, the final result is: \[ \boxed{0} \]