`int ((x ^(2) -x+1)/(x ^(2) +1)) e ^(cot^(-1) (x))dx =f (x) .e ^(cot ^(-1)(x)) +C`
where C is constant of integration. Then f (x) is equal to:

Options

A. `-x`
B. `sqrt (1-x)`
C. x
D. `sqrt(1+x)`

Answer

To solve the integral

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

we will follow these steps:

### Step 1: Substitution
Let \( t = \cot^{-1}(x) \). Then, we have:

\[ x = \cot(t) \quad \text{and} \quad dx = -\frac{1}{1 + x^2} \, dt = -\frac{1}{1 + \cot^2(t)} \, dt = -\frac{1}{\csc^2(t)} \, dt = -\sin^2(t) \, dt \]

### Step 2: Rewrite the integral
Now, we need to express \( x^2 - x + 1 \) and \( x^2 + 1 \) in terms of \( t \):

\[ x^2 = \cot^2(t), \quad x = \cot(t) \]

Thus,

\[ x^2 - x + 1 = \cot^2(t) - \cot(t) + 1 \]
\[ x^2 + 1 = \cot^2(t) + 1 = \csc^2(t) \]

Now substituting these into the integral gives:

\[ \int \frac{\cot^2(t) - \cot(t) + 1}{\csc^2(t)} e^t (-\sin^2(t)) \, dt \]

### Step 3: Simplify the integrand
We know that \( \csc^2(t) = \frac{1}{\sin^2(t)} \), so:

\[ \frac{\cot^2(t) - \cot(t) + 1}{\csc^2(t)} = (\cot^2(t) - \cot(t) + 1) \sin^2(t) \]

This simplifies to:

\[ (\cot^2(t) - \cot(t) + 1) \sin^2(t) = \cos^2(t) - \cos(t) \sin(t) + \sin^2(t) \]

### Step 4: Integrate
Now we can write the integral as:

\[ -\int (1 - \cot(t) + \sin^2(t)) e^t \, dt \]

This can be separated into three integrals:

\[ -\int e^t \, dt + \int \cot(t) e^t \, dt - \int \sin^2(t) e^t \, dt \]

### Step 5: Use Integration by Parts
For the integral involving \( \cot(t) e^t \), we will use integration by parts:

Let \( u = \cot(t) \) and \( dv = e^t dt \).

Then \( du = -\csc^2(t) dt \) and \( v = e^t \).

Applying integration by parts gives:

\[ \int \cot(t) e^t \, dt = \cot(t) e^t - \int e^t (-\csc^2(t)) \, dt \]

### Step 6: Substitute back
After evaluating the integrals, we substitute back \( t = \cot^{-1}(x) \) to express everything in terms of \( x \).

### Final Result
After all substitutions and simplifications, we find that:

\[ f(x) = x \]

Thus, the final answer is:

\[ f(x) = x \]