Answer is: option1

\(-4\)

Solution:

We are tasked with finding the value of:

\[ \int_1^3 f'(x) g(x) \, dx \]

Given that:

\[ \int_1^3 f(x) g'(x) \, dx = 8 \]

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:

\[ \int_1^3 f(x) g'(x) \, dx = \left[ f(x) g(x) \right]_1^3 - \int_1^3 f'(x) g(x) \, dx \]

Step 2: Substitute Given Information

We are given that:

\[ \int_1^3 f(x) g'(x) \, dx = 8 \]

Thus, the equation becomes:

\[ 8 = \left[ f(x) g(x) \right]_1^3 - \int_1^3 f'(x) g(x) \, dx \]

Step 3: Evaluate \( \left[ f(x) g(x) \right]_1^3 \)

From the table, we have the following values:

• At \( x = 1 \), \( f(1) = -2 \) and \( g(1) = 3 \)
• At \( x = 3 \), \( f(3) = 2 \) and \( g(3) = -1 \)

Now, calculate:

\[ \left[ f(x) g(x) \right]_1^3 = f(3) g(3) - f(1) g(1) = (2)(-1) - (-2)(3) = -2 + 6 = 4 \]

Step 4: Solve for the Unknown Integral

Substitute \( \left[ f(x) g(x) \right]_1^3 = 4 \) into the equation:

\[ 8 = 4 - \int_1^3 f'(x) g(x) \, dx \]

Solving for \( \int_1^3 f'(x) g(x) \, dx \):

\[ \int_1^3 f'(x) g(x) \, dx = 4 - 8 = -4 \]

Final Answer

The value of the integral is:

\[ \int_1^3 f'(x) g(x) \, dx = -4 \]