`int_2^4 (sqrt((x^2 - 4))/(4)) dx = `

Options

A. `2(3sqrt(3) - pi)//3`
B. `pi`
C. `2(3sqrt(3) - pi)`
D. None of these

Answer

To solve the integral \( \int_2^4 \frac{\sqrt{x^2 - 4}}{4} \, dx \), we can follow these steps:

### Step 1: Factor out the constant
Since \( \frac{1}{4} \) is a constant, we can factor it out of the integral:
\[ \int_2^4 \frac{\sqrt{x^2 - 4}}{4} \, dx = \frac{1}{4} \int_2^4 \sqrt{x^2 - 4} \, dx \]

**Hint:** Remember that constants can be factored out of integrals.

### Step 2: Use the integration formula
We will use the formula for the integral of the form \( \int \sqrt{x^2 - a^2} \, dx \):
\[ \int \sqrt{x^2 - a^2} \, dx = \frac{x}{2} \sqrt{x^2 - a^2} + \frac{a^2}{2} \log \left( x + \sqrt{x^2 - a^2} \right) + C \]
In our case, \( a^2 = 4 \) so \( a = 2 \).

### Step 3: Apply the formula
Now we apply the formula:
\[ \int \sqrt{x^2 - 4} \, dx = \frac{x}{2} \sqrt{x^2 - 4} + \frac{4}{2} \log \left( x + \sqrt{x^2 - 4} \right) + C \]
This simplifies to:
\[ \int \sqrt{x^2 - 4} \, dx = \frac{x}{2} \sqrt{x^2 - 4} + 2 \log \left( x + \sqrt{x^2 - 4} \right) + C \]

### Step 4: Evaluate the definite integral
Now we need to evaluate this from 2 to 4:
\[ \frac{1}{4} \left[ \left( \frac{x}{2} \sqrt{x^2 - 4} + 2 \log \left( x + \sqrt{x^2 - 4} \right) \right) \bigg|_2^4 \right] \]

### Step 5: Calculate the upper limit (x = 4)
Substituting \( x = 4 \):
\[ \frac{4}{2} \sqrt{4^2 - 4} + 2 \log \left( 4 + \sqrt{4^2 - 4} \right) = 2 \sqrt{12} + 2 \log \left( 4 + \sqrt{12} \right) \]
This simplifies to:
\[ 2 \sqrt{12} + 2 \log \left( 4 + 2\sqrt{3} \right) \]

### Step 6: Calculate the lower limit (x = 2)
Substituting \( x = 2 \):
\[ \frac{2}{2} \sqrt{2^2 - 4} + 2 \log \left( 2 + \sqrt{2^2 - 4} \right) = 0 + 2 \log(2) \]

### Step 7: Combine the results
Now we combine the results from the upper and lower limits:
\[ \frac{1}{4} \left[ \left( 2\sqrt{12} + 2 \log(4 + 2\sqrt{3}) \right) - 2 \log(2) \right] \]
This simplifies to:
\[ \frac{1}{4} \left( 2\sqrt{12} + 2 \log(4 + 2\sqrt{3}) - 2 \log(2) \right) \]
Factoring out the 2 gives:
\[ \frac{1}{2} \left( \sqrt{12} + \log \left( \frac{4 + 2\sqrt{3}}{2} \right) \right) \]

### Final Result
Thus, the final result of the integral is:
\[ \frac{1}{2} \left( \sqrt{12} + \log(2 + \sqrt{3}) \right) \]