https://r2.erweima.ai/mathbot/image/1725851269764wyutbmlx.jpg

The expression in the image appears to be:

\[
\int \frac{3x^{1/2}}{2(x^{3/2})} \, dx
\]

To solve this integral, let's simplify the integrand first:

\[
\frac{3x^{1/2}}{2x^{3/2}} = \frac{3}{2} \cdot \frac{x^{1/2}}{x^{3/2}} = \frac{3}{2} \cdot x^{(1/2 - 3/2)} = \frac{3}{2} \cdot x^{-1}
\]

Now the integral becomes:

\[
\int \frac{3}{2} \cdot x^{-1} \, dx
\]

The integral of \(x^{-1}\) is \(\ln|x|\), so the solution is:

\[
\frac{3}{2} \ln|x| + C
\]

where \(C\) is the constant of integration.

### Final Answer:
\[
\frac{3}{2} \ln|x| + C
\]

Do you need further clarification on any step or more details?

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Here are some related questions you might explore:
1. How do we handle integrals involving powers of \(x\)?
2. What are the rules for integrating functions of the form \(x^n\)?
3. What is the importance of the constant \(C\) in indefinite integrals?
4. How does the natural logarithm function arise in integrals?
5. Can you explain the relationship between exponents and logarithms in integrals?

**Tip**: Simplifying expressions before integrating often makes the process easier and helps avoid mistakes.

Solve the integral of (3x^(1/2)) / (2(x^(3/2))).

Learn how to solve the integral of (3x^(1/2)) / (2(x^(3/2))) with a detailed step-by-step explanation. This problem covers key calculus concepts, including exponent rules, logarithmic integration, and the power rule for integration. Suitable for students in Grades 11-12.

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