If `f(a-x)=f(x)` and `int_(0)^(a//2)f(x)dx=p`, then : `int_(0)^(a)f(x)dx=`

Options

A. 0
B. p
C. 2p
D. 3p

Answer

To solve the problem, we start with the given properties of the function \( f \) and the integral.

### Step 1: Understand the given property
We are given that \( f(a - x) = f(x) \). This means that the function \( f \) is symmetric about \( x = \frac{a}{2} \).

### Step 2: Set up the integral from 0 to a
We need to evaluate the integral \( \int_0^a f(x) \, dx \). We can split this integral into two parts:
\[ \int_0^a f(x) \, dx = \int_0^{\frac{a}{2}} f(x) \, dx + \int_{\frac{a}{2}}^a f(x) \, dx \]

### Step 3: Change of variable in the second integral
For the second integral, we can use the substitution \( x = a - t \). When \( x = \frac{a}{2} \), \( t = \frac{a}{2} \) and when \( x = a \), \( t = 0 \). Thus, we have:
\[ \int_{\frac{a}{2}}^a f(x) \, dx = \int_{\frac{a}{2}}^0 f(a - t) \, (-dt) = \int_0^{\frac{a}{2}} f(a - t) \, dt \]
Using the property \( f(a - t) = f(t) \), we can rewrite this as:
\[ \int_0^{\frac{a}{2}} f(a - t) \, dt = \int_0^{\frac{a}{2}} f(t) \, dt \]

### Step 4: Combine the integrals
Now we can combine the two integrals:
\[ \int_0^a f(x) \, dx = \int_0^{\frac{a}{2}} f(x) \, dx + \int_0^{\frac{a}{2}} f(x) \, dx = 2 \int_0^{\frac{a}{2}} f(x) \, dx \]
Since we know that \( \int_0^{\frac{a}{2}} f(x) \, dx = p \), we can substitute this into our equation:
\[ \int_0^a f(x) \, dx = 2p \]

### Final Answer
Thus, the value of the integral \( \int_0^a f(x) \, dx \) is:
\[ \int_0^a f(x) \, dx = 2p \]

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