If f (x) is the anti-derivative of `tan^(-1) sqrt(x)` such that the curve y = f(x) passes through the point (0,2) then f(x) =

Options

A. `x tan^(-1) sqrt(x) - sqrt(x) + tan^(-1) sqrt(x) + 2`
B. `x tan^(-1) sqrt(x) + sqrt(x) + tan^(-1) sqrt(x) + 2`
C. `x tan^(-1) sqrt(x) + 2`
D. None of these

Answer

To find the function \( f(x) \) which is the anti-derivative of \( \tan^{-1}(\sqrt{x}) \) and passes through the point \( (0, 2) \), we will follow these steps:

### Step 1: Set up the integral
We need to find the integral of \( \tan^{-1}(\sqrt{x}) \):
\[ f(x) = \int \tan^{-1}(\sqrt{x}) \, dx \]

### Step 2: Use integration by parts
To solve this integral, we will use integration by parts. We set:
- \( u = \tan^{-1}(\sqrt{x}) \)
- \( dv = dx \)

Then we find \( du \) and \( v \):
- \( du = \frac{1}{1 + x} \cdot \frac{1}{2\sqrt{x}} \, dx = \frac{1}{2\sqrt{x}(1+x)} \, dx \)
- \( v = x \)

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):
\[ f(x) = x \tan^{-1}(\sqrt{x}) - \int x \cdot \frac{1}{2\sqrt{x}(1+x)} \, dx \]

### Step 3: Simplify the remaining integral
The integral can be simplified:
\[ \int x \cdot \frac{1}{2\sqrt{x}(1+x)} \, dx = \int \frac{\sqrt{x}}{2(1+x)} \, dx \]

### Step 4: Substitute \( t = \sqrt{x} \)
Let \( t = \sqrt{x} \), then \( x = t^2 \) and \( dx = 2t \, dt \):
\[ \int \frac{t}{2(1+t^2)} \cdot 2t \, dt = \int \frac{t^2}{1+t^2} \, dt \]

### Step 5: Split the integral
We can split this integral:
\[ \int \frac{t^2}{1+t^2} \, dt = \int \left(1 - \frac{1}{1+t^2}\right) dt = \int dt - \int \frac{1}{1+t^2} \, dt \]
\[ = t - \tan^{-1}(t) + C \]

### Step 6: Substitute back for \( x \)
Now substituting back \( t = \sqrt{x} \):
\[ = \sqrt{x} - \tan^{-1}(\sqrt{x}) + C \]

### Step 7: Combine results
Now we can combine the results:
\[ f(x) = x \tan^{-1}(\sqrt{x}) - \left(\sqrt{x} - \tan^{-1}(\sqrt{x})\right) + C \]
\[ = x \tan^{-1}(\sqrt{x}) - \sqrt{x} + 2\tan^{-1}(\sqrt{x}) + C \]

### Step 8: Use the initial condition
We know that \( f(0) = 2 \):
\[ f(0) = 0 \cdot \tan^{-1}(0) - 0 + 2 \tan^{-1}(0) + C = 2 \]
This implies \( C = 2 \).

### Final result
Thus, the function \( f(x) \) is:
\[ f(x) = x \tan^{-1}(\sqrt{x}) - \sqrt{x} + 2\tan^{-1}(\sqrt{x}) + 2 \]