Find the value of integral $\int_{C}(x+y+z) d s$, where $C$ is parmeterized by $\vec{r}(t)=\langle 5 \sin (5 t), 4 \cos (5 t), 3 \cos (5 t)\rangle$ for $0 \leq t \leq \pi$.

Step 1 ：Let's parameterize the curve C by \(\vec{r}(t)=\langle 5 \sin (5 t), 4 \cos (5 t), 3 \cos (5 t)\rangle\) for \(0 \leq t \leq \pi\).

Step 2 ：Substitute x, y, and z in the integrand with their corresponding parameterized forms, we get \(x = 5\sin(5t)\), \(y = 4\cos(5t)\), and \(z = 3\cos(5t)\).

Step 3 ：Calculate the differentials dx, dy, and dz, we get \(dx = 25\cos(5t)\), \(dy = -20\sin(5t)\), and \(dz = -15\sin(5t)\).

Step 4 ：Calculate the differential ds, we get \(ds = \sqrt{625\sin(5t)^2 + 625\cos(5t)^2}\).

Step 5 ：Substitute these values into the integral and integrate over the given interval of t, we get the integral value is 50.

Step 6 ：\(\boxed{50}\) is the final answer.