Question

Let ∫-7to-1 f(x)dx=9,∫-7to-5f(x)dx=9,∫-3to-1f9x)dx=10.
Find ∫-5to-3 f(x)dx=?
and ∫-3to-5 (9f(x)-9)dx=?

Answer 1

Just plug in the values. If F(x) = ∫f(x) dx, then you have

∫[-7,-1] f(x)dx=9
∫[-7,-5] f(x)dx=9
∫[-3,-1] f(x)dx=10

So, that means
F(-1)-F(-7) = 9
F(-5)-F(-7) = 9
F(-1)-F(-3) = 10

From above, we see that F(-5) = F(-1)
∫[-5,-3] f(x) dx = F(-3) - F(-5)
= F(-3)-F(-1)
= -10

∫[-3,-5] (9f(x)-9)dx
= -∫[-5,-3] (9f(x)-9)dx
= -9∫[-5,-3] f(x) dx + 9∫[-5,-3] dx
= -9(-10) + 9x [-5,-3]
= 90 +9(-3+5)
= 90 + 18
= 108