`int_3^4 (dx)/sqrt(x^2+4)`

Answer

To solve the integral \( \int_3^4 \frac{dx}{\sqrt{x^2 + 4}} \), we will follow these steps:

### Step 1: Identify the Integral
We start with the integral:
\[ I = \int_3^4 \frac{dx}{\sqrt{x^2 + 4}} \]

### Step 2: Recognize the Form
The integral can be recognized as a standard form:
\[ \int \frac{dx}{\sqrt{x^2 + a^2}} = \ln |x + \sqrt{x^2 + a^2}| + C \]
where \( a = 2 \) in our case.

### Step 3: Apply the Formula
Using the formula, we can rewrite our integral:
\[ I = \left[ \ln |x + \sqrt{x^2 + 4}| \right]_3^4 \]

### Step 4: Evaluate at the Upper Limit
First, we evaluate at the upper limit \( x = 4 \):
\[ \ln |4 + \sqrt{4^2 + 4}| = \ln |4 + \sqrt{16 + 4}| = \ln |4 + \sqrt{20}| = \ln |4 + 2\sqrt{5}| \]

### Step 5: Evaluate at the Lower Limit
Now, we evaluate at the lower limit \( x = 3 \):
\[ \ln |3 + \sqrt{3^2 + 4}| = \ln |3 + \sqrt{9 + 4}| = \ln |3 + \sqrt{13}| \]

### Step 6: Combine the Results
Now we can combine the results from the upper and lower limits:
\[ I = \ln |4 + 2\sqrt{5}| - \ln |3 + \sqrt{13}| \]

### Step 7: Use Logarithmic Properties
Using the property of logarithms \( \ln a - \ln b = \ln \frac{a}{b} \):
\[ I = \ln \left( \frac{4 + 2\sqrt{5}}{3 + \sqrt{13}} \right) \]

### Final Answer
Thus, the value of the integral is:
\[ \int_3^4 \frac{dx}{\sqrt{x^2 + 4}} = \ln \left( \frac{4 + 2\sqrt{5}}{3 + \sqrt{13}} \right) \]

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