Answer is: option4

18

Solution:

We are tasked with finding the value of the following integral:

\[ \int_0^3 x f''(x) \, dx \]

Step 1: Use Integration by Parts

We will use the integration by parts formula:

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

Let us set:

\[ u = x \quad \Rightarrow \quad du = dx \]

\[ dv = f''(x) \, dx \quad \Rightarrow \quad v = f'(x) \]

Applying integration by parts to \( \int x f''(x) \, dx \), we get:

\[ \int_0^3 x f''(x) \, dx = \left[ x f'(x) \right]_0^3 - \int_0^3 f'(x) \, dx \]

Step 2: Evaluate \( \left[ x f'(x) \right]_0^3 \)

From the table, we have the following values:

• At \( x = 0 \), \( f'(0) = -2 \)
• At \( x = 3 \), \( f'(3) = 7 \)

Now, evaluate \( \left[ x f'(x) \right]_0^3 \):

\[ \left[ x f'(x) \right]_0^3 = (3)(7) - (0)(-2) = 21 \]

Step 3: Evaluate \( \int_0^3 f'(x) \, dx \)

The integral of \( f'(x) \) is \( f(x) \). Using the values from the table:

• At \( x = 0 \), \( f(0) = 2 \)
• At \( x = 3 \), \( f(3) = 5 \)

Thus, we have:

\[ \int_0^3 f'(x) \, dx = f(3) - f(0) = 5 - 2 = 3 \]

Step 4: Final Calculation

Now, substitute everything back into the formula:

\[ \int_0^3 x f''(x) \, dx = 21 - 3 = 18 \]

Final Answer

The value of the integral is:

\[ \int_0^3 x f''(x) \, dx = 18 \]