The integers between $\sqrt{19}$ and $\sqrt{90}$ are determined by identifying the smallest integer greater than $\sqrt{19}$ and the largest integer less than $\sqrt{90}$. Since $\sqrt{19}$ is approximately 4.36, the first integer greater than it is 5. Meanwhile, $\sqrt{90}$ is approximately 9.49, making the largest integer less than it equal to 9. Therefore, the integers between these two values are 5, 6, 7, 8, and 9. In total, there are five integers between $\sqrt{19}$ and $\sqrt{90}$.