$$ \frac{e^{x^{2}}.\sin (3x-1)}{\ln (4x-1)+3} $$

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Certainly! Here's the mathematical expression you provided, rendered for clarity:

$$ 
\frac{e^{x^{2}} \cdot \sin(3x - 1)}{\ln(4x - 1) + 3} 
$$

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### Breakdown of the Expression

Let's take a closer look at each component of the expression:

1. **Numerator:**
   - **Exponential Function:** $$ e^{x^{2}} $$
     - This represents the exponential function with the exponent $$ x^2 $$.
   - **Sine Function:** $$ \sin(3x - 1) $$
     - This is the sine of a linear function of $$ x $$, specifically $$ 3x - 1 $$.

2. **Denominator:**
   - **Natural Logarithm:** $$ \ln(4x - 1) $$
     - This is the natural logarithm of the linear function $$ 4x - 1 $$.
   - **Constant Addition:** $$ + 3 $$
     - This adds 3 to the logarithmic term.

### Domain Considerations

For the expression to be defined, certain conditions must be met:

1. **Inside the Logarithm:**
   - The argument of the natural logarithm must be positive:
     $$
     4x - 1 > 0 \implies x > \frac{1}{4}
     $$
   
2. **Denominator Non-Zero:**
   - The entire denominator cannot be zero:
     $$
     \ln(4x - 1) + 3 \neq 0 \implies \ln(4x - 1) \neq -3 \implies 4x - 1 \neq e^{-3} \implies x \neq \frac{1 + e^{-3}}{4}
     $$
   
   - **Approximate Value:** $$ x \neq \frac{1 + e^{-3}}{4} \approx 0.275 $$

**Combined Domain:**
$$
x > \frac{1}{4} \quad \text{and} \quad x \neq \frac{1 + e^{-3}}{4}
$$

### Potential Operations

Depending on your needs, here are some operations you might consider performing on this function:

1. **Differentiation:**
   - To find the derivative $$ f'(x) $$ of the function:
     $$
     f(x) = \frac{e^{x^{2}} \cdot \sin(3x - 1)}{\ln(4x - 1) + 3}
     $$
     You would apply the **Quotient Rule**:
     $$
     f'(x) = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}
     $$
     Where:
     - $$ f(x) = e^{x^{2}} \cdot \sin(3x - 1) $$
     - $$ g(x) = \ln(4x - 1) + 3 $$

2. **Integration:**
   - Finding the integral of this function might be complex and may not have a closed-form solution. Numerical methods or approximation techniques might be necessary.

3. **Graphing:**
   - Visualizing the function within its domain can provide insights into its behavior, such as intercepts, asymptotes, and periodicity.

4. **Simplification:**
   - Given the complexity of the function, simplification may be limited. However, certain trigonometric identities or logarithmic properties could be applied depending on the context.

### Let Me Know How I Can Help Further!

If you have a specific operation or analysis you'd like to perform on this expression—such as finding its derivative, evaluating it at a particular point, or graphing it—please let me know, and I'd be happy to assist!
The expression is: $$ \frac{e^{x^{2}} \cdot \sin(3x - 1)}{\ln(4x - 1) + 3} $$ **Domain:** $$ x > \frac{1}{4} $$ and $$ x \neq \frac{1 + e^{-3}}{4} $$

\( \frac{e^{x^{2}}.\sin (3x-1)}{\ln (4x-1)+3} \)

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