$$ \int_{}^{}\!\frac{e^{\sqrt{x}}}{\sqrt{x}}\,\mathrm{d}x $$

The given integral is:

\[
\int \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx
\]

To solve this, we can apply the substitution method. Let's set:

\[
u = \sqrt{x} \quad \text{so that} \quad x = u^2 \quad \text{and} \quad dx = 2u \, du
\]

Now substitute $ u $ and $ dx $ into the integral:

\[
\int \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx = \int \frac{e^u}{u} \cdot 2u \, du = 2 \int e^u \, du
\]

The integral of $ e^u $ is simply $ e^u $, so we have:

\[
2 \int e^u \, du = 2e^u + C
\]

Finally, substitute back $ u = \sqrt{x} $ to get the solution in terms of $ x $:

\[
2e^u + C = 2e^{\sqrt{x}} + C
\]

Thus, the solution to the integral is:

\[
\boxed{2e^{\sqrt{x}} + C}
\]

Would you like further details or clarification?

Here are 5 related questions to explore further:
1. How would you solve $\int e^{x^2} \, dx$?
2. What is the significance of substitution in integrals involving square roots?
3. Can you derive a similar solution for $\int e^{x^{1/3}} \, dx$?
4. What happens if the exponential is in the denominator, e.g. $\int \frac{1}{e^{\sqrt{x}}} \, dx$?
5. How would you handle definite integrals with limits for this problem?

**Tip:** When dealing with integrals involving roots or powers, substitution often simplifies the process greatly by reducing the expression to a more familiar form.

$$ \int \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx $$

Explore how to solve the integral $$ \int \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx $$ using the substitution method. This detailed solution covers essential calculus concepts, including exponential functions and the substitution theorem. Ideal for students in Grades 11-12.

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