Evaluate: `int_(-1)^4f(x)dx=4a n dint_2^4(3-f(x))dx=7,` then find the value of `int_2^(-1)f(x)dxdot`

Explanation

To solve the problem step by step, we start with the given information:

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

2. We can simplify the second integral:
\[ \int_{2}^{4} (3 - f(x)) \, dx = \int_{2}^{4} 3 \, dx - \int_{2}^{4} f(x) \, dx \]
The integral of a constant is simply the constant multiplied by the length of the interval:
\[ \int_{2}^{4} 3 \, dx = 3 \cdot (4 - 2) = 3 \cdot 2 = 6 \]
Therefore, we can write:
\[ 6 - \int_{2}^{4} f(x) \, dx = 7 \]

3. Rearranging this gives us:
\[ -\int_{2}^{4} f(x) \, dx = 7 - 6 \]
\[ -\int_{2}^{4} f(x) \, dx = 1 \]
Thus:
\[ \int_{2}^{4} f(x) \, dx = -1 \]

4. Now, we can use the result from step 1:
\[ \int_{-1}^{4} f(x) \, dx = \int_{-1}^{2} f(x) \, dx + \int_{2}^{4} f(x) \, dx \]
Substituting the known values:
\[ 4 = \int_{-1}^{2} f(x) \, dx + (-1) \]
This simplifies to:
\[ \int_{-1}^{2} f(x) \, dx = 4 + 1 = 5 \]

5. Finally, we need to find:
\[ \int_{2}^{-1} f(x) \, dx \]
We can use the property of integrals that states:
\[ \int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx \]
Thus:
\[ \int_{2}^{-1} f(x) \, dx = -\int_{-1}^{2} f(x) \, dx = -5 \]

So, the final answer is:
\[ \int_{2}^{-1} f(x) \, dx = -5 \]