We are tasked with finding the integral of the function:

$\int_4^9 \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx$

Let's proceed step-by-step to solve this.

Step 1: Substitution

We'll use substitution to simplify the integral. Let:

$u = \sqrt{x} \quad \Rightarrow \quad x = u^2$

Now differentiate $x$ with respect to $u$:

$dx = 2u \, du$

Step 2: Change of limits

When $x = 4$, $u = \sqrt{4} = 2$. When $x = 9$, $u = \sqrt{9} = 3$.

Thus, the limits change from $x = 4$ to $x = 9$ to $u = 2$ to $u = 3$.

Step 3: Substituting into the integral

Now substitute everything into the original integral:

$\int_4^9 \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx = \int_2^3 \frac{e^u}{u} \cdot 2u \, du$

The $u$ terms cancel out, leaving:

$= 2 \int_2^3 e^u \, du$

Step 4: Integration

The integral of $e^u$ is simply $e^u$. Therefore, we can compute:

$2 \int_2^3 e^u \, du = 2 \left[ e^u \right]_2^3$

Step 5: Applying the limits

Now, substitute the limits into the expression:

$= 2 \left( e^3 - e^2 \right)$

Thus, the value of the integral is:

$2(e^3 - e^2)$

This is the final answer.

Do you want further details or have any questions?

Here are some related questions to explore:

1. How does substitution work in integration, and why was it used here?
2. What is the general rule for integrating exponential functions?
3. How do you apply the Fundamental Theorem of Calculus in definite integrals?
4. What other substitution techniques could be useful for integrals involving square roots?
5. How do integrals involving natural exponentials compare to logarithmic integrals?

Tip: Always verify the limits of integration carefully when performing substitution in definite integrals!