What is `int(dx)/(sqrt(4+x^(2)))` equal to ?

Options

A. `log|sqrt(4+x^(2))+x|+C`
B. `log|sqrt(4+x^(2))-x|+C`
C. `sin^(-1)((x)/(2))+C`
D. None of these

Answer

To solve the integral \( \int \frac{dx}{\sqrt{4+x^2}} \), we can use a known formula for the integral of the form \( \int \frac{dx}{\sqrt{a^2 + x^2}} \).

### Step-by-step Solution:

1. **Identify the form**:
We recognize that our integral is of the form \( \int \frac{dx}{\sqrt{a^2 + x^2}} \) where \( a^2 = 4 \). Thus, \( a = 2 \).

2. **Apply the formula**:
The formula for this integral is:
\[ \int \frac{dx}{\sqrt{a^2 + x^2}} = \ln |x + \sqrt{x^2 + a^2}| + C \]
Substituting \( a = 2 \) into the formula gives:
\[ \int \frac{dx}{\sqrt{4 + x^2}} = \ln |x + \sqrt{x^2 + 4}| + C \]

3. **Final result**:
Therefore, the integral \( \int \frac{dx}{\sqrt{4+x^2}} \) is equal to:
\[ \ln |x + \sqrt{x^2 + 4}| + C \]