I gather that the given functions satisfy the following definite integral relations:
[tex]\displaystyle \int_{-1}^4 f(x) = -6[/tex]
[tex]\displaystyle \int_4^5 f(x) = 10[/tex]
[tex]\displaystyle \int_4^5 g(x) =5[/tex]
a) By linearity of the integral operator, we have
[tex]\displaystyle \int_{-1}^4 10 f(x) \, dx = 10 \int_{-1}^4 f(x) \, dx = 10 \times (-6) = \boxed{-60}[/tex]
b) The integral over an interval is equal to the sum of integrals over a partition of that interval. In this case, the interval [-1, 5] can be written as the interval union [-1, 4] U [4, 5], so that
[tex]\displaystyle \int_{-1}^5 f(x) \, dx = \int_{-1}^4 f(x) \, dx + \int_4^5 f(x) \, dx = -6 + 10 = \boxed{4}[/tex]
c) By linearity,
[tex]\displaystyle \int_4^5 (f(x) - g(x)) \, dx = \int_4^5 f(x) \, dx - \int_4^5 g(x) \, dx = 10 - 5 = \boxed{5}[/tex]
d) By linearity,
[tex]\displaystyle \int_4^5 (4f(x) + 5g(x)) \, dx = 4 \int_4^5 f(x) \, dx + 5 \int_4^5 g(x) \, dx = 4\times10 + 5\times5 = \boxed{65}[/tex]
