Let F(x) be an indefinite integral of `sin^(2)x`
Statement-1: The function F(x) satisfies `F(x+pi)=F(x)` for all real x. because
Statement-2: `sin^(3)(x+pi)=sin^(2)x` for all real x.
A) Statement-1: True , statement-2 is true, Statement -2 is not a correct explanation for statement -1
c) Statement-1 is True, Statement -2 is False.
D) Statement-1 is False, Statement-2 is True.

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 is false, Statement II is true.

Answer

To solve the problem, we need to analyze the two statements provided and determine their validity.

### Step 1: Understanding Statement 1
Statement 1 claims that the function \( F(x) \) satisfies the equation:
\[ F(x + \pi) = F(x) \quad \text{for all real } x. \]

### Step 2: Finding the Indefinite Integral
The function \( F(x) \) is defined as the indefinite integral of \( \sin^2 x \):
\[ F(x) = \int \sin^2 x \, dx. \]

### Step 3: Periodicity of \( \sin^2 x \)
The function \( \sin^2 x \) has a period of \( \pi \). This means:
\[ \sin^2(x + \pi) = \sin^2 x \quad \text{for all real } x. \]

### Step 4: Evaluating the Integral Over One Period
Since \( \sin^2 x \) is periodic with period \( \pi \), we can conclude that:
\[ F(x + \pi) = \int \sin^2(x + \pi) \, dx = \int \sin^2 x \, dx = F(x). \]
Thus, Statement 1 is **True**.

### Step 5: Understanding Statement 2
Statement 2 claims:
\[ \sin^3(x + \pi) = \sin^2 x \quad \text{for all real } x. \]

### Step 6: Evaluating \( \sin^3(x + \pi) \)
Using the property of sine:
\[ \sin(x + \pi) = -\sin x, \]
we can evaluate \( \sin^3(x + \pi) \):
\[ \sin^3(x + \pi) = (-\sin x)^3 = -\sin^3 x. \]

### Step 7: Comparing \( \sin^3(x + \pi) \) with \( \sin^2 x \)
Clearly, we have:
\[ \sin^3(x + \pi) = -\sin^3 x \neq \sin^2 x. \]
Thus, Statement 2 is **False**.

### Conclusion
- Statement 1 is True.
- Statement 2 is False.

The correct option is **C**: Statement-1 is True, Statement-2 is False.