`intsqrt(1+x^(2)) dx` is equal to

Options

A. `(x)/(2)sqrt(1+x^(2)) + (1)/(2) log |(x + sqrt(1 + x^(2)))| + C`
B. `(2)/(3)(1+x^(2))^(3//2) + C`
C. `(2)/(3)x(1+x^(2))^(3//2) + C`
D. `(x^(2))/(2) sqrt(1 + x^(2)) + (1)/(2)x^(2)log (x + sqrt(1 + x^(2))) + C`

Answer

To solve the integral \( \int \sqrt{1 + x^2} \, dx \), we can use a standard formula for integrating expressions of the form \( \sqrt{a^2 + x^2} \).

### Step-by-Step Solution:

1. **Identify the Formula**:
The formula for integrating \( \sqrt{a^2 + x^2} \) is given by:
\[ \int \sqrt{a^2 + x^2} \, dx = \frac{x}{2} \sqrt{a^2 + x^2} + \frac{a^2}{2} \ln \left| x + \sqrt{a^2 + x^2} \right| + C \]
where \( C \) is the constant of integration.

2. **Substituting Values**:
In our case, we have \( a = 1 \). Therefore, we substitute \( a = 1 \) into the formula:
\[ \int \sqrt{1 + x^2} \, dx = \frac{x}{2} \sqrt{1 + x^2} + \frac{1^2}{2} \ln \left| x + \sqrt{1 + x^2} \right| + C \]

3. **Simplifying the Expression**:
Now, simplify the expression:
\[ = \frac{x}{2} \sqrt{1 + x^2} + \frac{1}{2} \ln \left| x + \sqrt{1 + x^2} \right| + C \]

4. **Final Result**:
Thus, the integral \( \int \sqrt{1 + x^2} \, dx \) is:
\[ \int \sqrt{1 + x^2} \, dx = \frac{x}{2} \sqrt{1 + x^2} + \frac{1}{2} \ln \left| x + \sqrt{1 + x^2} \right| + C \]