Suppose that f(x) is an even function and let the integral f(x) dx from 0 to 1=5 and the integral f(x) dx from 0 to 7 =1. what is the integral from f(x) dx from -7 to -1?

Since it's even, the integral from -7 to -1 is the same as the integral from 1 to 7.
think about
x^2 from 0..1,its 1/3
from 0..7 its 7^3/3
from -1..7 its 7^3/3+1/3

from -7..-1 of x^2 its -1/3+7^3/3 to get this without integrating we just do 7^3/3-1/3 which is the integral from 0..7 minus the integral from 0..1
So the integral from 1 to 0 f(x) dx = 5 changes to the integral from 0 to 1 f(x) dx = -5 and you add it to the integral from 0 to 7 which is 1 making -4

2/3 because 8/12 divided down is 4/6 which reduced again is 2/3