Verified answer

I guess you're saying that

the integral of f(x) dx from x = -7 to -1 is 1;

the integral of f(x) dx from x = -10 to -5 is 18;

the integral of f(x) dx from x = -10 to -1 is 9.

And they want to know the integral of f(x) dx from x = -7 to -5.

Not very hard actually. Let "F(x)" be the indefinite integral of f(x) dx, that is to say, f(x) is the derivative of F(x). Then we know that

F(-1) - F(-7) = 1,

F(-5) - F(-10) = 18,

F(-1) - F(-10) = 9.

Subtracting eqn 3 from eqn 2 gives:

F(-5) - F(-1) = 9.

Adding that to eqn 1 gives:

F(-5) - F(-7) = 10.

So that's the answer. 10.

int[ f(x)] dx from - 5 to -7 = int [f(x)] dx from -5 to - 10 + int[ f(x)] dx from -10 to - 1 + int[ f(x)] dx from - 1 to - 7 = - 18 + 9 - 1 = - 10

** int [ f(x)] from - 5 to - 10 = - int[ f(x)] dx from - 10 to - 5 = - 18

** int [f(x)] dx from - 1 to - 7 = - int{ f(x)] dx from - 7 to - 1 = - 1

∫−10(top)−5(bottom) f(x) dx+∫−1(top)−10(bottom) f(x) dx+∫−7(top)−1(bottom) f(x) dx=-18+9-1=-10