If `f(x)=int(x^(2)+sin^(2)x)/(1+x^(2))sec^(2)xdx and f(0)=0,` then `f(1)=`

Options

A. `1 - (pi)/(4)`
B. `(pi)/(4) -1`
C. `tan 1- (pi)/(4)`
D. None of these

Answer

To solve the problem, we need to evaluate the integral given by the function \( f(x) \) and then find \( f(1) \) given that \( f(0) = 0 \).

### Step-by-step Solution:

1. **Write down the function:**
\[ f(x) = \int \frac{x^2 + \sin^2 x}{1 + x^2} \sec^2 x \, dx \]

2. **Substitute \( \sin^2 x \):**
We know that \( \sin^2 x = 1 - \cos^2 x \). So we can rewrite the integral:
\[ f(x) = \int \frac{x^2 + (1 - \cos^2 x)}{1 + x^2} \sec^2 x \, dx \]
This simplifies to:
\[ f(x) = \int \frac{x^2 + 1 - \cos^2 x}{1 + x^2} \sec^2 x \, dx \]

3. **Split the integral:**
We can separate the integral into two parts:
\[ f(x) = \int \frac{x^2 + 1}{1 + x^2} \sec^2 x \, dx - \int \frac{\cos^2 x}{1 + x^2} \sec^2 x \, dx \]
The first part simplifies since \( \sec^2 x = \frac{1}{\cos^2 x} \):
\[ f(x) = \int \sec^2 x \, dx - \int \frac{\cos^2 x}{1 + x^2} \sec^2 x \, dx \]

4. **Integrate the first term:**
The integral of \( \sec^2 x \) is:
\[ \int \sec^2 x \, dx = \tan x + C \]

5. **Evaluate the second integral:**
The second integral can be simplified:
\[ \int \frac{\cos^2 x}{1 + x^2} \sec^2 x \, dx = \int \frac{1}{1 + x^2} \, dx = \tan^{-1}(x) + C \]

6. **Combine the results:**
Thus, we have:
\[ f(x) = \tan x - \tan^{-1}(x) + C \]

7. **Use the condition \( f(0) = 0 \):**
Evaluating at \( x = 0 \):
\[ f(0) = \tan(0) - \tan^{-1}(0) + C = 0 - 0 + C = C \]
Since \( f(0) = 0 \), we find \( C = 0 \).

8. **Final expression for \( f(x) \):**
Therefore, we have:
\[ f(x) = \tan x - \tan^{-1}(x) \]

9. **Evaluate \( f(1) \):**
Now we calculate \( f(1) \):
\[ f(1) = \tan(1) - \tan^{-1}(1) \]
We know \( \tan^{-1}(1) = \frac{\pi}{4} \):
\[ f(1) = \tan(1) - \frac{\pi}{4} \]

10. **Final result:**
Thus, the final answer is:
\[ f(1) = \tan(1) - \frac{\pi}{4} \]