\(\int \sqrt{x} \, dx\) solution in latex-code

Here is the solution to the integral \(\int \sqrt{x} \, dx\) in LaTeX code format:

```latex
\documentclass{article}
\usepackage{amsmath}

\begin{document}

To solve the integral \(\int \sqrt{x} \, dx\), we first rewrite the square root in terms of an exponent:

\[
\int \sqrt{x} \, dx = \int x^{1/2} \, dx
\]

Using the power rule for integration, which states that

\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for} \, n \neq -1,
\]

we can apply it to our integral:

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

Simplifying, we multiply the numerator and the denominator by 2 to get:

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

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.

\end{document}
```

This code will render the solution to the integral \(\int \sqrt{x} \, dx\) in a clear and formatted way, using the LaTeX document preparation system. If you compile this code using a LaTeX editor, it will generate a well-formatted document with the solution steps clearly presented.

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Do you need any further help with this, or would you like more details on any specific part? Here are some additional questions and topics related to this integral:

1. How do you derive the power rule for integration?
2. What are the steps for integrating more complex functions?
3. How can LaTeX be used to format other types of mathematical expressions?
4. What is the role of the constant of integration in indefinite integrals?
5. How do you check the correctness of an integration result?
6. What happens if the power \(n = -1\) in the power rule?
7. How does integration relate to finding the area under a curve?
8. Can you perform integration using numerical methods?

**Tip:** Learning to use LaTeX effectively can greatly enhance the presentation of mathematical work and is widely used in academic publications.

Explore the solution to the integral \( \int \sqrt{x} \, dx \) step-by-step. Learn how to apply the power rule for integration and understand the role of the constant of integration. Suitable for advanced high school and college students.

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