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The image contains a mathematical expression that represents an integral. The second integral shown is:

\[
\int \frac{dx}{\sqrt{3 + 3x^2}}
\]

To solve this integral, we can proceed as follows:

1. **Factor the expression under the square root:**

   \[
   \sqrt{3 + 3x^2} = \sqrt{3(1 + x^2)} = \sqrt{3} \cdot \sqrt{1 + x^2}
   \]

2. **Simplify the integral:**

   \[
   \int \frac{dx}{\sqrt{3 + 3x^2}} = \frac{1}{\sqrt{3}} \int \frac{dx}{\sqrt{1 + x^2}}
   \]

3. **Recognize the standard form:**

   The integral \(\int \frac{dx}{\sqrt{1 + x^2}}\) is a standard form, which equals \(\text{arcsinh}(x)\) or \(\ln(x + \sqrt{x^2 + 1})\).

4. **Write the solution:**

   \[
   \int \frac{dx}{\sqrt{1 + x^2}} = \ln\left|x + \sqrt{x^2 + 1}\right| + C
   \]

   Therefore, the solution to the original integral is:

   \[
   \frac{1}{\sqrt{3}} \ln\left|x + \sqrt{x^2 + 1}\right| + C
   \]

Let me know if you would like further clarification or more details on the solution steps!

Here are 5 related questions that might deepen your understanding:

1. How do you recognize standard integrals like \(\int \frac{dx}{\sqrt{1 + x^2}}\)?
2. What are some techniques to simplify integrals involving square roots?
3. Can the arcsinh function be used instead of logarithms for this solution?
4. How would you solve this integral using trigonometric substitution?
5. What is the geometric meaning of the integral \(\int \frac{dx}{\sqrt{1 + x^2}}\)?

**Tip:** Familiarize yourself with standard integrals, such as those involving square roots of quadratics, as they appear frequently in calculus.

This page provides a detailed solution to the integral \( \int \frac{dx}{\sqrt{3 + 3x^2}} \), including algebraic manipulation and the use of standard integral formulas. Suitable for advanced calculus students.

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