If `int_(-1) ^(4) f(x) dx=4and int_(2)^(4) (3-f(x))dx=7,` then
`int_(-1) ^(2) f(x) dx=`

Options

A. `-2`
B. 3
C. 4
D. 5

Answer

To solve the problem, we need to find the value of the integral \(\int_{-1}^{2} f(x) \, dx\) given the following information:

1. \(\int_{-1}^{4} f(x) \, dx = 4\)
2. \(\int_{2}^{4} (3 - f(x)) \, dx = 7\)

Let's break down the steps to find \(\int_{-1}^{2} f(x) \, dx\).

### Step 1: Evaluate the second integral
We start with the second integral:

\[ \int_{2}^{4} (3 - f(x)) \, dx = 7 \]

This can be split into two separate integrals:

\[ \int_{2}^{4} 3 \, dx - \int_{2}^{4} f(x) \, dx = 7 \]

### Step 2: Calculate \(\int_{2}^{4} 3 \, dx\)
Now, we calculate \(\int_{2}^{4} 3 \, dx\):

\[ \int_{2}^{4} 3 \, dx = 3 \cdot (4 - 2) = 3 \cdot 2 = 6 \]

### Step 3: Substitute back into the equation
Substituting this value back into our equation from Step 1 gives:

\[ 6 - \int_{2}^{4} f(x) \, dx = 7 \]

### Step 4: Solve for \(\int_{2}^{4} f(x) \, dx\)
Rearranging the equation to solve for \(\int_{2}^{4} f(x) \, dx\):

\[ -\int_{2}^{4} f(x) \, dx = 7 - 6 \]
\[ -\int_{2}^{4} f(x) \, dx = 1 \]
\[ \int_{2}^{4} f(x) \, dx = -1 \]

### Step 5: Use the first integral
Now, we have:

\[ \int_{-1}^{4} f(x) \, dx = 4 \]

We can express this integral as:

\[ \int_{-1}^{4} f(x) \, dx = \int_{-1}^{2} f(x) \, dx + \int_{2}^{4} f(x) \, dx \]

### Step 6: Substitute known values
Substituting the known value of \(\int_{2}^{4} f(x) \, dx\):

\[ 4 = \int_{-1}^{2} f(x) \, dx + (-1) \]

### Step 7: Solve for \(\int_{-1}^{2} f(x) \, dx\)
Now we solve for \(\int_{-1}^{2} f(x) \, dx\):

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

### Final Answer
Thus, the value of \(\int_{-1}^{2} f(x) \, dx\) is:

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