Question

Let the integral ∫ from 4 to 13 f(x) dx = 5, ∫ from 4 to 7 f(x) dx = 1, ∫ from 10 to 13 f(x) dx = 3.

a)Find ∫ from 7 to 10 f(x)dx= ?
and
b)Find ∫ from 10 to 7 [5 f(x)−1]dx= ?

Answer 1

∫[4,13] = ∫[4,7] + ∫[7,10] + ∫[10,13]
so,
5 = 1 + ∫[7,10] + 3
so, ∫[7,10] = 1

Now, you know that
∫[7,10] 5f(x)-1 dx = 5∫[7,10] f(x) dx - ∫[7,10] 1 dx
= 5*1 - 3 = 2
Now, recall that ∫[10,7] = -∫[7,10] = -2

Answer 2

Where does the -3 come from in 5*1 - 3 = 2 ? I do understand the rest of your steps though

Answer 3

just evaluate ∫[7,10] 1 dx = x [7,10] = 10-7 = 3