Let f(x) be an indefinite integral of `sin^(2)x`. Consider the following statements :
Statements
1. The function f(x) satisfies `f(x+pi)=f(x)` for all real x.
2. `Sin^(2)(x+pi)=sin^(2)x` for all real x.
Which one of the following is correct in respect of the above statements ?

Options

A. Both the statements are true and statement 2 is the correct explanation of statement 1
B. Both the statements are true but statement 2 not the correct explanation of statement 1
C. Statement 1 is true but statement 2 is false
D. Statement 1 is false but statement 2 is true

Answer

To solve the problem, we need to analyze the two statements given about the function \( f(x) \), which is an indefinite integral of \( \sin^2 x \).

### Step 1: Find the indefinite integral of \( \sin^2 x \)

We start with the integral:
\[ f(x) = \int \sin^2 x \, dx \]

Using the identity:
\[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]
we can rewrite the integral as:
\[ f(x) = \int \frac{1 - \cos(2x)}{2} \, dx \]

### Step 2: Separate the integral

Now, we can separate the integral:
\[ f(x) = \frac{1}{2} \int 1 \, dx - \frac{1}{2} \int \cos(2x) \, dx \]

### Step 3: Integrate each part

Calculating the first integral:
\[ \int 1 \, dx = x \]
Calculating the second integral:
\[ \int \cos(2x) \, dx = \frac{1}{2} \sin(2x) \]
Thus, we have:
\[ f(x) = \frac{1}{2} x - \frac{1}{2} \cdot \frac{1}{2} \sin(2x) + C \]
which simplifies to:
\[ f(x) = \frac{x}{2} - \frac{1}{4} \sin(2x) + C \]

### Step 4: Verify the first statement \( f(x + \pi) = f(x) \)

Now, we need to check if:
\[ f(x + \pi) = f(x) \]

Calculating \( f(x + \pi) \):
\[ f(x + \pi) = \frac{x + \pi}{2} - \frac{1}{4} \sin(2(x + \pi)) + C \]
Using the sine function property \( \sin(2(x + \pi)) = \sin(2x + 2\pi) = \sin(2x) \):
\[ f(x + \pi) = \frac{x + \pi}{2} - \frac{1}{4} \sin(2x) + C \]
This simplifies to:
\[ f(x + \pi) = \frac{x}{2} + \frac{\pi}{2} - \frac{1}{4} \sin(2x) + C \]

Since \( f(x) = \frac{x}{2} - \frac{1}{4} \sin(2x) + C \), we can see that:
\[ f(x + \pi) \neq f(x) \]
Thus, the first statement is **false**.

### Step 5: Verify the second statement \( \sin^2(x + \pi) = \sin^2 x \)

Using the periodic property of sine:
\[ \sin(x + \pi) = -\sin x \]
Thus:
\[ \sin^2(x + \pi) = (-\sin x)^2 = \sin^2 x \]
This shows that the second statement is **true**.

### Conclusion

- **Statement 1**: False
- **Statement 2**: True

The correct answer is that **Statement 1 is false and Statement 2 is true**.