If `f(0) = 1 , f(2) = 3, f'(2) = 5` and `f'(0)` is finite, then `int_(0)^(1)x. f^''(2x) dx` is equal to

Answer

To solve the problem, we need to evaluate the integral:

\[ \int_{0}^{1} x \cdot f''(2x) \, dx \]

Given the information:
- \( f(0) = 1 \)
- \( f(2) = 3 \)
- \( f'(2) = 5 \)
- \( f'(0) \) is finite.

### Step 1: Apply Integration by Parts

We will use the integration by parts formula:

\[ \int u \, dv = uv - \int v \, du \]

Let:
- \( u = x \) (thus \( du = dx \))
- \( dv = f''(2x) \, dx \) (thus \( v = \int f''(2x) \, dx \))

To find \( v \), we need to integrate \( f''(2x) \):

\[ v = \int f''(2x) \, dx = \frac{1}{2} f'(2x) + C \]

### Step 2: Substitute into Integration by Parts

Now substituting \( u \) and \( v \) into the integration by parts formula:

\[ \int_{0}^{1} x f''(2x) \, dx = \left[ x \cdot \frac{1}{2} f'(2x) \right]_{0}^{1} - \int_{0}^{1} \frac{1}{2} f'(2x) \, dx \]

### Step 3: Evaluate the Boundary Terms

Calculating the boundary term:

\[ \left[ x \cdot \frac{1}{2} f'(2x) \right]_{0}^{1} = \left(1 \cdot \frac{1}{2} f'(2)\right) - \left(0 \cdot \frac{1}{2} f'(0)\right) = \frac{1}{2} f'(2) - 0 = \frac{1}{2} \cdot 5 = \frac{5}{2} \]

### Step 4: Evaluate the Remaining Integral

Now we need to evaluate:

\[ \int_{0}^{1} \frac{1}{2} f'(2x) \, dx \]

Using the substitution \( u = 2x \), then \( du = 2dx \) or \( dx = \frac{du}{2} \). The limits change from \( x = 0 \) to \( x = 1 \) which corresponds to \( u = 0 \) to \( u = 2 \).

Thus, we have:

\[ \int_{0}^{1} \frac{1}{2} f'(2x) \, dx = \frac{1}{2} \int_{0}^{2} f'(u) \cdot \frac{du}{2} = \frac{1}{4} \int_{0}^{2} f'(u) \, du \]

### Step 5: Evaluate the Integral of \( f' \)

Using the Fundamental Theorem of Calculus:

\[ \int_{0}^{2} f'(u) \, du = f(2) - f(0) = 3 - 1 = 2 \]

Thus:

\[ \frac{1}{4} \int_{0}^{2} f'(u) \, du = \frac{1}{4} \cdot 2 = \frac{1}{2} \]

### Step 6: Combine Results

Now substituting back into our integration by parts result:

\[ \int_{0}^{1} x f''(2x) \, dx = \frac{5}{2} - \frac{1}{2} = \frac{4}{2} = 2 \]

### Final Answer

Thus, the value of the integral is:

\[ \int_{0}^{1} x f''(2x) \, dx = 2 \]