Statement I Integral of an even function is not always an odd function. Statement II Integral of an odd function is an even function .

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 analyze the statements given in the question, we will break down the definitions of even and odd functions, and then evaluate the validity of each statement step by step.

### Step 1: Understanding Even and Odd Functions
- **Even Function**: A function \( f(x) \) is called even if \( f(-x) = f(x) \) for all \( x \) in the domain of \( f \). Examples include \( f(x) = x^2 \) and \( f(x) = \cos(x) \).
- **Odd Function**: A function \( f(x) \) is called odd if \( f(-x) = -f(x) \) for all \( x \) in the domain of \( f \). Examples include \( f(x) = x^3 \) and \( f(x) = \sin(x) \).

### Step 2: Evaluating Statement I
**Statement I**: "Integral of an even function is not always an odd function."
- Let’s consider an even function, for example, \( f(x) = x^2 \).
- The integral of \( f(x) \) is:
\[ \int f(x) \, dx = \int x^2 \, dx = \frac{x^3}{3} + C \]
- The function \( \frac{x^3}{3} + C \) is neither even nor odd because:
- \( f(-x) = \frac{(-x)^3}{3} + C = -\frac{x^3}{3} + C \) which does not equal \( f(x) \) or \( -f(x) \).
- Therefore, Statement I is **true**.

### Step 3: Evaluating Statement II
**Statement II**: "Integral of an odd function is an even function."
- Let’s consider an odd function, for example, \( f(x) = x^3 \).
- The integral of \( f(x) \) is:
\[ \int f(x) \, dx = \int x^3 \, dx = \frac{x^4}{4} + C \]
- The function \( \frac{x^4}{4} + C \) is even because:
- \( f(-x) = \frac{(-x)^4}{4} + C = \frac{x^4}{4} + C \) which equals \( f(x) \).
- Therefore, Statement II is **true**.

### Conclusion
- **Statement I** is true: The integral of an even function is not always an odd function.
- **Statement II** is true: The integral of an odd function is indeed an even function.

### Final Answer
Both statements are true, but Statement II is not a correct explanation for Statement I.

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