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Veseljchak [2.6K]
2 years ago
14

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

Mathematics
1 answer:
anygoal [31]2 years ago
7 0

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 \times (-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\times10 + 5\times5 = \boxed{65}

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