Question

∫3,−2 f(x)dx = −4 and ∫4,3 f(x)dx = 3.
Find ∫3,−2 (−4f(x)+3) dx = ______

Answer 1

I am new to integrals but i assumed that since f(x)dx is -4 and the integral is the same value you can just plug in -4 for f(x) and solve but it was incorrect

Answer 2

You cannot plug in f(x) = 4.
All you know is that ∫f(x) dx = 4

∫(-4f(x)+3) dx = ∫-4f(x) dx + ∫3 dx
= -4∫f(x) dx + ∫3 dx
= -4∫f(x) dx + 3x
Now, using the limits [-2,3], that is
= -4(-4) + (3*3 - 3(-2))
= 16 + 15
= 31