Answer is: option3

13

Solution:

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

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

We are given that:

\[ \int_0^3 f'(x) g(x) \, dx = 6 \]

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:

\[ u = f(x) \quad \Rightarrow \quad du = f'(x) \, dx \]

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

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

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

Step 2: Substitute Given Information

We are given that:

\[ \int_0^3 f'(x) g(x) \, dx = 6 \]

Thus, the equation becomes:

\[ \int_0^3 f(x) g'(x) \, dx = \left[ f(x) g(x) \right]_0^3 - 6 \]

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

From the table, we have the following values:

• At \( x = 0 \), \( f(0) = 1 \) and \( g(0) = -4 \)
• At \( x = 3 \), \( f(3) = 5 \) and \( g(3) = 3 \)

Now, calculate:

\[ \left[ f(x) g(x) \right]_0^3 = f(3) g(3) - f(0) g(0) = (5)(3) - (1)(-4) = 15 + 4 = 19 \]

Step 4: Solve for the Unknown Integral

Substitute \( \left[ f(x) g(x) \right]_0^3 = 19 \) into the equation:

\[ \int_0^3 f(x) g'(x) \, dx = 19 - 6 = 13 \]

Final Answer

The value of the integral is:

\[ \int_0^3 f(x) g'(x) \, dx = 13 \]