Question

Evaluate the definite integrals using properties of the definite integral and the fact that

∫
4
− 1
f(x) dx = − 6, ∫
5
4
f(x) dx = 10, and ∫
5
4
g(x) dx = 5.
(a) ∫
4
− 1
10f(x) dx = ____________
(b) ∫
5
− 1
f(x) dx = ____________
(c) ∫
5
4
(f(x) − g(x)) dx = ____________
(d) ∫
5
4
(4f(x) + 5g(x)) dx = ___________

Answer 1

I gather that the given functions satisfy the following definite integral relations:

`\displaystyle \int_(-1)^4 f(x) = -6`

`\displaystyle \int_4^5 f(x) = 10`

`\displaystyle \int_4^5 g(x) =5`

a) By linearity of the integral operator, we have

`\displaystyle \int_(-1)^4 10 f(x) \, dx = 10 \int_(-1)^4 f(x) \, dx = 10 * (-6) = \boxed{-60}`

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

`\displaystyle \int_(-1)^5 f(x) \, dx = \int_(-1)^4 f(x) \, dx + \int_4^5 f(x) \, dx = -6 + 10 = \boxed{4}`

c) By linearity,

`\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}`

d) By linearity,

`\displaystyle \int_4^5 (4f(x) + 5g(x)) \, dx = 4 \int_4^5 f(x) \, dx + 5 \int_4^5 g(x) \, dx = 4*10 + 5*5 = \boxed{65}`