To solve the integral $\int \sqrt{x} \, dx$, we can use the power rule for integration. The function $\sqrt{x}$ can be rewritten in exponential form as $x^{1/2}$. The power rule for integration states that:

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

where $n \neq -1$ and $C$ is the constant of integration. Applying this rule to our integral:

$\int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} + C = \frac{x^{3/2}}{3/2} + C$

To simplify further, we can multiply the numerator and the denominator by $2$ to eliminate the fraction in the denominator:

$= \frac{2}{3} x^{3/2} + C$

Thus, the integral of $\sqrt{x}$ is:

$\int \sqrt{x} \, dx = \frac{2}{3} x^{3/2} + C$

where $C$ is the constant of integration.

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Do you want more details or have any questions about this solution? Here are some related questions:

1. What is the power rule for integration?
2. How do you integrate functions with fractional exponents?
3. What is the constant of integration, and why is it important?
4. How can you verify the result of an indefinite integral?
5. How do definite integrals differ from indefinite integrals?
6. How can you use substitution to solve more complex integrals?
7. What are common mistakes to avoid when integrating functions with roots?
8. How do you handle integration when the power is negative?

Tip: Always rewrite roots and radicals as fractional exponents to simplify integration.