\( \int ^ { \pi } 2 ^ { 4 } \cdot \cos ^ { 8 } \frac { x } { x } d x \)

To evaluate the integral \( \int ^ { \pi } 2^{4} \cdot \cos^{8} \frac{x}{x} \, dx \), we first simplify it. Notably, since \(\cos^{8} \frac{x}{x} = \cos^{8}(1)\) for every \(x \neq 0\), this means the integral simplifies to evaluating a constant function: \[ \int ^{ \pi } 2^{4} \cdot \cos^{8}(1) \, dx = 2^{4} \cdot \cos^{8}(1) \cdot \int^{\pi} dx \] Calculating the integral \( \int^{\pi} dx \) gives \( \pi \). Therefore, the expression becomes: \[ 2^{4} \cdot \cos^{8}(1) \cdot \pi = 16 \cdot \cos^{8}(1) \cdot \pi \] So the final result is: \[ \int ^{ \pi } 2^{4} \cdot \cos^{8} \frac{x}{x} \, dx = 16 \pi \cdot \cos^{8}(1) \] This is how we can evaluate the integral intuitively by recognizing the behavior of the integral with respect to its limits and the nature of the function involved.