Need help on calculus problem

Question

Let f be a function such that  and . What is the value of ?

O 12

O 14

O 16

O 18

Answer 1

Some properties of integrals that are useful:

∫^b_{a f(x) dx = - ∫}^a_b f(x) dx

∫^c_a f(x) dx = ∫^b_a f(x) dx + ∫^c_b f(x) dx if a < b < c

So if:

∫¹₅ f(x) dx = 1, then ∫⁵₁ f(x) dx = -1

and if:

∫¹_-1 f(x) dx = 5,

then ∫⁵_-1 f(x) dx = ∫¹_-1 f(x) dx + ∫⁵₁ f(x) dx

Substitute:

∫⁵_-1 f(x) dx = 5 - 1 = 4

multiply 3 on both sides:

3 ∫⁵_-1 f(x) dx = 12