We are tasked with finding the equation of the tangent line to the graph of the given function $F(x)$ at $x = 1$. The function is defined as:

$F(x) = \int_{\sqrt{x}}^{x} \tan(t) \, dt$

Solution Steps:

1. Differentiate $F(x)$ using the Fundamental Theorem of Calculus: The function $F(x)$ involves both the upper and lower limits as functions of $x$. To differentiate, we apply the Leibniz rule:

   $F'(x) = \tan(x) \cdot \frac{d}{dx}(x) - \tan(\sqrt{x}) \cdot \frac{d}{dx}(\sqrt{x})$

   This simplifies to:

   $F'(x) = \tan(x) - \tan(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}$

2. Evaluate $F'(x)$ at $x = 1$: Substitute $x = 1$ into $F'(x)$:

   $F'(1) = \tan(1) - \tan(\sqrt{1}) \cdot \frac{1}{2\sqrt{1}}$

   Since $\sqrt{1} = 1$, this becomes:

   $F'(1) = \tan(1) - \tan(1) \cdot \frac{1}{2}$

   Simplify further:

   $F'(1) = \tan(1) \left(1 - \frac{1}{2}\right) = \frac{\tan(1)}{2}$

3. Find $F(1)$: To calculate $F(1)$, evaluate the definite integral:

   $F(1) = \int_{\sqrt{1}}^{1} \tan(t) \, dt$

   Since the lower and upper limits are the same ($\sqrt{1} = 1$), the value of the integral is:

   $F(1) = 0$

4. Equation of the tangent line: The equation of the tangent line at $x = 1$ is:

   $y - F(1) = F'(1)(x - 1)$

   Substitute $F(1) = 0$ and $F'(1) = \frac{\tan(1)}{2}$:

   $y = \frac{\tan(1)}{2}(x - 1)$

Final Answer:

The equation of the tangent line is:

$y = \frac{\tan(1)}{2}(x - 1)$

Would you like me to elaborate further or compute approximations for $\tan(1)$?

Here are some related questions for further exploration:

1. How do we evaluate the integral of trigonometric functions for different limits?
2. What is the geometric interpretation of $F(x)$ as a function involving definite integrals?
3. How does the Leibniz rule apply when the limits of integration depend on $x$?
4. What is the significance of the slope of the tangent line in the context of $F(x)$?
5. Can we derive higher-order derivatives of $F(x)$ using similar techniques?

Tip: Always verify your differentiation steps when working with integral bounds that are functions of $x$.