If f(x) is periodic function with period, T, then

Options

A. `int_(a)^(b) f (x) dx=int_(a)^(b+T)f(x) dx`
B. `int_(a)^(b)f(x)=int_(a+T)^(b+T)f(x)dx`
C. `int_(a)^(b)f(x) dx=int_(a+T)^(b)f(x)dx`
D. `int_(a)^(b)f(x) dx=int_(a+T)^(b+2T)f(x)dx`

Answer

To solve the problem, we need to analyze the properties of a periodic function \( f(x) \) with period \( T \). A periodic function satisfies the condition \( f(x + T) = f(x) \) for all \( x \). We will evaluate the given options to determine which one is correct.

### Step 1: Understanding the Periodicity
A periodic function \( f(x) \) with period \( T \) means:
\[ f(x + T) = f(x) \]
This property will be crucial in evaluating the integrals.

### Step 2: Evaluate Option 1
We need to check if:
\[ \int_a^b f(x) \, dx = \int_a^{b + T} f(x) \, dx \]
Using the periodicity:
\[ \int_a^{b + T} f(x) \, dx = \int_a^b f(x) \, dx + \int_b^{b + T} f(x) \, dx \]
Since \( f(x) \) is periodic:
\[ \int_b^{b + T} f(x) \, dx = \int_b^{b + T} f(x) \, dx = \int_0^T f(x) \, dx \]
Thus, the left-hand side does not equal the right-hand side, so option 1 is incorrect.

### Step 3: Evaluate Option 2
Now, we check if:
\[ \int_a^b f(x) \, dx = \int_a^{b + T} f(x) \, dx \]
As shown in Step 2, the integral from \( a \) to \( b + T \) can be expressed as:
\[ \int_a^b f(x) \, dx + \int_b^{b + T} f(x) \, dx \]
Since \( \int_b^{b + T} f(x) \, dx \) is not zero, option 2 is also incorrect.

### Step 4: Evaluate Option 3
Next, we check:
\[ \int_a^b f(x) \, dx = \int_a^{b + 2T} f(x) \, dx \]
Using the periodicity:
\[ \int_a^{b + 2T} f(x) \, dx = \int_a^b f(x) \, dx + \int_b^{b + T} f(x) \, dx + \int_{b + T}^{b + 2T} f(x) \, dx \]
Both \( \int_b^{b + T} f(x) \, dx \) and \( \int_{b + T}^{b + 2T} f(x) \, dx \) are equal to \( \int_0^T f(x) \, dx \), so option 3 is incorrect.

### Step 5: Evaluate Option 4
Finally, we check:
\[ \int_a^b f(x) \, dx = \int_a^{b + 2T} f(x) \, dx \]
As shown previously, the left-hand side does not equal the right-hand side, making option 4 incorrect.

### Conclusion
After evaluating all options, we find that none of the options satisfy the condition for periodic functions. However, if we were to find a correct statement, it would be that:
\[ \int_a^{b + T} f(x) \, dx = \int_a^b f(x) \, dx + \int_b^{b + T} f(x) \, dx \]
This shows the integral over one period can be added to the integral over the interval \( [a, b] \).