Let `int(dx)/(sqrt(x^(2)+1)-x)=f(x)+C` such that `f(0)=0` and C is the constant of integration, then the value of `f(1)` is

Options

A. `(1)/(sqrt2)+(1)/(2)ln(1+sqrt2)`
B. `(1)/(2)+(1)/(sqrt2)ln(1+sqrt2)`
C. `(1)/(2)+(1)/(2)lnsqrt2+1`
D. `(1)/(sqrt2)+(1)/(2)(1+ln(1+sqrt2))`

Answer

To solve the problem, we need to evaluate the integral:

\[ \int \frac{dx}{\sqrt{x^2 + 1} - x} = f(x) + C \]

where \( f(0) = 0 \) and \( C \) is the constant of integration. We are tasked with finding \( f(1) \).

### Step 1: Rationalize the Denominator

To simplify the integral, we can rationalize the denominator:

\[ \frac{1}{\sqrt{x^2 + 1} - x} \cdot \frac{\sqrt{x^2 + 1} + x}{\sqrt{x^2 + 1} + x} = \frac{\sqrt{x^2 + 1} + x}{(\sqrt{x^2 + 1})^2 - x^2} \]

The denominator simplifies as follows:

\[ (\sqrt{x^2 + 1})^2 - x^2 = x^2 + 1 - x^2 = 1 \]

Thus, we have:

\[ \int \frac{dx}{\sqrt{x^2 + 1} - x} = \int (\sqrt{x^2 + 1} + x) \, dx \]

### Step 2: Integrate the Expression

Now we can integrate:

\[ \int (\sqrt{x^2 + 1} + x) \, dx = \int \sqrt{x^2 + 1} \, dx + \int x \, dx \]

The second integral is straightforward:

\[ \int x \, dx = \frac{x^2}{2} \]

For the first integral, we can use the formula for the integral of \( \sqrt{x^2 + a^2} \):

\[ \int \sqrt{x^2 + 1} \, dx = \frac{x}{2} \sqrt{x^2 + 1} + \frac{1}{2} \ln |x + \sqrt{x^2 + 1}| \]

Combining these results, we have:

\[ \int (\sqrt{x^2 + 1} + x) \, dx = \frac{x}{2} \sqrt{x^2 + 1} + \frac{1}{2} \ln |x + \sqrt{x^2 + 1}| + \frac{x^2}{2} + C \]

### Step 3: Define \( f(x) \)

Thus, we can express \( f(x) \) as:

\[ f(x) = \frac{x}{2} \sqrt{x^2 + 1} + \frac{1}{2} \ln |x + \sqrt{x^2 + 1}| + \frac{x^2}{2} \]

### Step 4: Determine the Constant \( C \)

We were given that \( f(0) = 0 \):

\[ f(0) = \frac{0}{2} \sqrt{0^2 + 1} + \frac{1}{2} \ln |0 + \sqrt{0^2 + 1}| + \frac{0^2}{2} + C = 0 \]

This simplifies to:

\[ 0 + \frac{1}{2} \ln(1) + 0 + C = 0 \]

Since \( \ln(1) = 0 \), we find \( C = 0 \).

### Step 5: Calculate \( f(1) \)

Now we need to find \( f(1) \):

\[ f(1) = \frac{1}{2} \sqrt{1^2 + 1} + \frac{1}{2} \ln |1 + \sqrt{1^2 + 1}| + \frac{1^2}{2} \]

Calculating each term:

1. \( \frac{1}{2} \sqrt{2} = \frac{\sqrt{2}}{2} \)
2. \( \frac{1}{2} \ln(1 + \sqrt{2}) \)
3. \( \frac{1}{2} \)

Putting it all together:

\[ f(1) = \frac{\sqrt{2}}{2} + \frac{1}{2} \ln(1 + \sqrt{2}) + \frac{1}{2} \]

Combining the constant terms:

\[ f(1) = \frac{\sqrt{2}}{2} + \frac{1}{2} + \frac{1}{2} \ln(1 + \sqrt{2}) \]

### Final Answer

Thus, the value of \( f(1) \) is:

\[ f(1) = \frac{1 + \sqrt{2}}{2} + \frac{1}{2} \ln(1 + \sqrt{2}) \]