Answer is: option4
\( \frac{25}{2} \)Solution:
When you reverse the limits of an integral, the sign changes:
\[ \int_6^4 f(x)\,dx = -\int_4^6 f(x)\,dx = -5 \Rightarrow \int_4^6 f(x)\,dx = -5 \]
Notice that:
\[ \int_1^6 f(x)\,dx = \int_1^4 f(x)\,dx + \int_4^6 f(x)\,dx \]
Substitute the known values:
\[ \frac{15}{2} = \int_1^4 f(x)\,dx + (-5) \]
Solve for \( \int_1^4 f(x)\,dx \):
\[ \int_1^4 f(x)\,dx = \frac{15}{2} + 5 = \frac{15}{2} + \frac{10}{2} = \frac{25}{2} \]
Option D