Two coherent sources of light with intensities I_1 and I_2 produce...

The intensity at any point in an interference pattern is given by $I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$, where $\phi$ is the phase difference between the two waves.

To find the average intensity over the entire screen, we take the average of the intensity expression over a full cycle of phase difference. The average value of $\cos\phi$ over a complete cycle is zero.

Therefore, $I_{\text{avg}} = I_1 + I_2 + 2\sqrt{I_1 I_2}(0) = I_1 + I_2$.

This relation shows that the total energy distributed over the screen is exactly equal to the sum of the individual energies from the two sources, proving that energy is conserved and merely redistributed.

Answer: $I_{\text{avg}} = I_1 + I_2$