Answer is: option3

Solution:

Given Information:

1. \(\int_{-2}^{6} f(x)\,dx = 10\)
2. \(\int_{2}^{6} f(x)\,dx = 3\)

Find \(\int_{-2}^{2} f(x)\,dx\):

\[ \int_{-2}^{6} f(x)\,dx = \int_{-2}^{2} f(x)\,dx + \int_{2}^{6} f(x)\,dx \] \[ 10 = \int_{-2}^{2} f(x)\,dx + 3 \] \[ \int_{-2}^{2} f(x)\,dx = 10 - 3 = 7 \]

Evaluate \(\int_{2}^{6} f(4 - x)\,dx\):

Let's perform a substitution: \( u = 4 - x \Rightarrow du = -dx \Rightarrow dx = -du \)

1. When \(x = 2\), \(u = 4 - 2 = 2\)
2. When \(x = 6\), \(u = 4 - 6 = -2\)

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

We already found that \(\int_{-2}^{2} f(u)\,du = 7\)

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

Option C