Find theintegral of exponential root x division root x dx in the limits of 9 and 4

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.

Find the integral of exponential root x division root x dx in the limits of 9 and 4

This step-by-step solution demonstrates how to solve the integral of e^√x divided by √x, using substitution methods. The final answer is given as 2(e^3 - e^2). Suitable for advanced high school or college-level students.

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