Statement I `int((1)/(1+x^(4)))dx=tan^(-1)(x^(2))+C`
Statement II `int(1)/(1+x^(2))dx=tan^(-1)x +C`

Options

A. Statement I is true, Statement II is also true , Statement II is the correct explanation of Statement I.
B. Statement I is true, Statement II is also true, Statement II is not the correct explanation of Statement I.
C. Statement I is true, Statement II is false.
D. Statement I is false, Statement II is true .

Answer

To determine the validity of the two statements regarding indefinite integrals, we will analyze each statement step by step.

### Statement I:
\[ \int \frac{1}{1+x^4} \, dx = \tan^{-1}(x^2) + C \]

**Step 1: Analyze the Integral**
The integral \(\int \frac{1}{1+x^4} \, dx\) does not have a straightforward antiderivative. We need to manipulate it to find a solution.

**Step 2: Factor the Denominator**
We can factor \(1 + x^4\) as:
\[ 1 + x^4 = (x^2 + \sqrt{2}x + 1)(x^2 - \sqrt{2}x + 1) \]
This allows us to use partial fraction decomposition.

**Step 3: Set Up Partial Fraction Decomposition**
We express:
\[ \frac{1}{1+x^4} = \frac{A}{x^2 + \sqrt{2}x + 1} + \frac{B}{x^2 - \sqrt{2}x + 1} \]
Multiplying through by the denominator gives:
\[ 1 = A(x^2 - \sqrt{2}x + 1) + B(x^2 + \sqrt{2}x + 1) \]

**Step 4: Solve for Coefficients A and B**
Expanding and equating coefficients, we can find values for \(A\) and \(B\). This will involve solving a system of equations.

**Step 5: Integrate Each Term**
Once we have \(A\) and \(B\), we can integrate each term separately. The integrals will involve arctangent functions.

**Step 6: Conclusion for Statement I**
After performing the integration, we find that the result does not simplify to \(\tan^{-1}(x^2) + C\). Therefore, **Statement I is false**.

### Statement II:
\[ \int \frac{1}{1+x^2} \, dx = \tan^{-1}(x) + C \]

**Step 1: Analyze the Integral**
The integral \(\int \frac{1}{1+x^2} \, dx\) is a standard integral.

**Step 2: Recognize the Standard Result**
The integral \(\int \frac{1}{1+x^2} \, dx\) is known to equal \(\tan^{-1}(x) + C\).

**Step 3: Conclusion for Statement II**
Since this is a well-known result, **Statement II is true**.

### Final Conclusion:
- **Statement I is false.**
- **Statement II is true.**