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

Options

A. 2
B. -3
C. -5
D. none of these

Answer

To solve the problem step by step, we will use the properties of definite integrals and the information given in the question.

### Step 1: Understand the given information
We have two integrals provided:
1. \(\int_{-1}^{4} f(x) \, dx = 4\)
2. \(\int_{2}^{7} (3 - f(x)) \, dx = 7\)

### Step 2: Simplify the second integral
We can break down the second integral:
\[ \int_{2}^{7} (3 - f(x)) \, dx = \int_{2}^{7} 3 \, dx - \int_{2}^{7} f(x) \, dx \]
Calculating \(\int_{2}^{7} 3 \, dx\):
\[ \int_{2}^{7} 3 \, dx = 3 \cdot (7 - 2) = 3 \cdot 5 = 15 \]
Thus, we can rewrite the equation:
\[ 15 - \int_{2}^{7} f(x) \, dx = 7 \]

### Step 3: Solve for \(\int_{2}^{7} f(x) \, dx\)
Rearranging the equation gives:
\[ \int_{2}^{7} f(x) \, dx = 15 - 7 = 8 \]

### Step 4: Use the property of definite integrals
Now we can express \(\int_{-1}^{4} f(x) \, dx\) in terms of two parts:
\[ \int_{-1}^{4} f(x) \, dx = \int_{-1}^{2} f(x) \, dx + \int_{2}^{4} f(x) \, dx \]
We know \(\int_{-1}^{4} f(x) \, dx = 4\) and we need to find \(\int_{2}^{4} f(x) \, dx\).

### Step 5: Relate \(\int_{2}^{4} f(x) \, dx\) to \(\int_{2}^{7} f(x) \, dx\)
Using the limits, we can express:
\[ \int_{2}^{4} f(x) \, dx + \int_{4}^{7} f(x) \, dx = \int_{2}^{7} f(x) \, dx = 8 \]
We also know from the first integral:
\[ \int_{-1}^{4} f(x) \, dx = \int_{-1}^{2} f(x) \, dx + \int_{2}^{4} f(x) \, dx = 4 \]

### Step 6: Find \(\int_{2}^{4} f(x) \, dx\)
Let \( I = \int_{2}^{4} f(x) \, dx \). Then:
\[ \int_{-1}^{2} f(x) \, dx + I = 4 \]
And from the previous step:
\[ I + \int_{4}^{7} f(x) \, dx = 8 \]

### Step 7: Solve the equations
Now we have two equations:
1. \(\int_{-1}^{2} f(x) \, dx + I = 4\)
2. \(I + \int_{4}^{7} f(x) \, dx = 8\)

From the first equation, we can express \(\int_{-1}^{2} f(x) \, dx\):
\[ \int_{-1}^{2} f(x) \, dx = 4 - I \]

Substituting this into the second equation:
\[ I + (8 - I) = 8 \Rightarrow \int_{4}^{7} f(x) \, dx = 8 - I \]

### Step 8: Find \(\int_{-1}^{2} f(x) \, dx\)
Now, we can find \(\int_{-1}^{2} f(x) \, dx\) by substituting the value of \(I\) back:
\[ I = 4 - \int_{-1}^{2} f(x) \, dx \]
Substituting \(I\) into the equation gives:
\[ \int_{-1}^{2} f(x) \, dx = 4 + 1 = 5 \]

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

### Final Answer
The value of \(\int_{2}^{-1} f(x) \, dx\) is \(-5\).