$$ \int\left(\cdot\right)=\sqrt{\cdot+3} $$

The equation you've written involves an integral that results in a square root function. Let's denote the integral as:

\[
\int f(x) \, dx = \sqrt{g(x) + 3}
\]

To solve this, we need to find the function $ f(x) $ such that when integrated, the result is $ \sqrt{g(x) + 3} $.

### Step 1: Differentiate the Result
Assume the antiderivative is given by:

\[
F(x) = \sqrt{g(x) + 3}
\]

The derivative of this with respect to $ x $ gives the integrand $ f(x) $. Using the chain rule, we differentiate:

\[
f(x) = \frac{d}{dx} \left( \sqrt{g(x) + 3} \right)
\]

This simplifies to:

\[
f(x) = \frac{1}{2\sqrt{g(x) + 3}} \cdot g'(x)
\]

Thus, the integrand $ f(x) $ must be of the form:

\[
f(x) = \frac{g'(x)}{2\sqrt{g(x) + 3}}
\]

Now, if you know the specific form of $ g(x) $, we can determine $ f(x) $ explicitly. Otherwise, this is the general form of the integrand.

Would you like more details or clarification on this process?

### Related Questions:
1. How do you compute integrals involving square roots?
2. What is the chain rule in calculus, and how is it applied here?
3. Can you provide a concrete example with a specific $ g(x) $?
4. What techniques are used to solve non-elementary integrals?
5. How do you determine $ g(x) $ from a given $ f(x) $?
6. What are the common forms of integrals that involve roots?
7. Can partial fractions be applied in similar cases?
8. How does substitution help in integrating complex functions?

**Tip:** Always differentiate your answer to check if your integration was correct.

Explore how to integrate the square root function ∫(·) = √(· + 3) step-by-step. Understand the application of the Fundamental Theorem of Calculus and the Chain Rule in solving this integral problem. Suitable for college-level calculus students.

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