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= ?

∫[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

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

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

To find the value of the integral from 7 to 10 of f(x)dx, we can use the property of linearity of integrals.

a) The integral from 4 to 13 of f(x)dx can be split into the sum of two integrals: from 4 to 7 and from 7 to 13. So, we can write:

∫ from 4 to 13 f(x)dx = ∫ from 4 to 7 f(x)dx + ∫ from 7 to 13 f(x)dx

Substituting the given values:

5 = 1 + ∫ from 7 to 13 f(x)dx

To find the value of the integral from 7 to 13 of f(x)dx, we subtract the known value of the integral from 7 to 13 of f(x)dx:

∫ from 7 to 13 f(x)dx = 5 - 1 = 4

Now, to find the value of the integral from 7 to 10 of f(x)dx, we can split the integral from 7 to 13 into two parts: from 7 to 10 and from 10 to 13. So, we can write:

∫ from 7 to 13 f(x)dx = ∫ from 7 to 10 f(x)dx + ∫ from 10 to 13 f(x)dx

Substituting the known values:

4 = ∫ from 7 to 10 f(x)dx + 3

To find the value of the integral from 7 to 10 of f(x)dx, we subtract the known value of the integral from 10 to 13 of f(x)dx:

∫ from 7 to 10 f(x)dx = 4 - 3 = 1

b) Now, let's consider the integral from 10 to 7 [5 f(x)−1]dx. We can rewrite this integral as:

∫ from 10 to 7 [5 f(x)−1]dx = -∫ from 7 to 10 [5 f(x)−1]dx

Since we found the value of the integral from 7 to 10 of f(x)dx as 1, we can substitute this back into the equation:

∫ from 10 to 7 [5 f(x)−1]dx = -1