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

Options

A. 2
B. `-3`
C. `-5`
D. 1

Answer

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

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

### Step 1: Rewrite the second integral
We can rewrite the second integral as:
\[ \int_{2}^{4} [3 - f(x)] \, dx = \int_{2}^{4} 3 \, dx - \int_{2}^{4} f(x) \, dx \]
Calculating \(\int_{2}^{4} 3 \, dx\):
\[ \int_{2}^{4} 3 \, dx = 3 \times (4 - 2) = 3 \times 2 = 6 \]
So we have:
\[ 6 - \int_{2}^{4} f(x) \, dx = 7 \]

### Step 2: Solve for \(\int_{2}^{4} f(x) \, dx\)
Rearranging the equation gives:
\[ -\int_{2}^{4} f(x) \, dx = 7 - 6 \]
\[ -\int_{2}^{4} f(x) \, dx = 1 \]
Thus,
\[ \int_{2}^{4} f(x) \, dx = -1 \]

### Step 3: Use the property of definite integrals
Now, we can use the property of definite integrals:
\[ \int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx \]
This means:
\[ \int_{2}^{-1} f(x) \, dx = -\int_{-1}^{2} f(x) \, dx \]

### Step 4: Break down the integral from \(-1\) to \(4\)
Using the known value of \(\int_{-1}^{4} f(x) \, dx\):
\[ \int_{-1}^{4} f(x) \, dx = \int_{-1}^{2} f(x) \, dx + \int_{2}^{4} f(x) \, dx \]
Substituting the known values:
\[ 4 = \int_{-1}^{2} f(x) \, dx + (-1) \]
Thus,
\[ \int_{-1}^{2} f(x) \, dx = 4 + 1 = 5 \]

### Step 5: Find \(\int_{2}^{-1} f(x) \, dx\)
Now, substituting back:
\[ \int_{2}^{-1} f(x) \, dx = -\int_{-1}^{2} f(x) \, dx = -5 \]

### Final Answer
Thus, the value of \(\int_{2}^{-1} f(x) \, dx\) is:
\[ \boxed{-5} \]