We are tasked with evaluating the following definite integral:

$I = \int_{4}^{9} \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx$

Step-by-step solution:

Step 1: Substitution

Let’s perform a substitution to simplify the expression.

Let $u = \sqrt{x}$, so: $x = u^2 \quad \text{and} \quad dx = 2u \, du$

Step 2: Change the limits of integration

When $x = 4$, we have $u = \sqrt{4} = 2$, and when $x = 9$, we have $u = \sqrt{9} = 3$.

Thus, the new limits of integration are from $u = 2$ to $u = 3$.

Step 3: Substituting into the integral

Now, we substitute $x = u^2$ and $dx = 2u \, du$ into the integral: $I = \int_{2}^{3} \frac{e^u}{u} \cdot 2u \, du$

The $u$-terms cancel out, leaving us with: $I = 2 \int_{2}^{3} e^u \, du$

Step 4: Integrate

The integral of $e^u$ is simply $e^u$. Therefore: $I = 2 \left[ e^u \right]_2^3$

Step 5: Evaluate the definite integral

Now, we evaluate the definite integral: $I = 2 \left( e^3 - e^2 \right)$

Thus, the value of the integral is: $I = 2 \left( e^3 - e^2 \right)$

Final Answer:

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

Would you like to calculate the numerical value, or need further clarification?

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Related Questions:

1. How do we choose substitutions in integrals?
2. What is the integral of $e^{x}$?
3. How do definite and indefinite integrals differ?
4. Can we always cancel variables in integration?
5. How are substitution methods applied in other integral types?

Tip:

When integrating functions involving square roots, try using substitution methods like $u = \sqrt{x}$ to simplify the expression.