Integral Help!! If f(x) is odd, find: int{x=-2 to x=2} f(x)dx =

Question

Let ∫₀² f(x) dx = 8

If f(x) is odd, find:

∫_-2² f(x)dx =

Answer 1

Since f(x) is odd, the graph of y = f(x) is symmetric with respect to the origin.

So, ∫_{(-2 to 2)}f(x)dx = ∫_{(-2 to 0)}f(x)dx + ∫_{(0 to 2)}f(x)dx = -8 + 8 = 0

Answer 2

∫-2 to 2 f(x) dx = ∫-2 to 0 f(x) dx + ∫0 to 2 f(x) dx = ∫0 to 2 f(-x) dx + ∫0 to 2 f(x) dx

As f(x) is odd, f(-x) = -f(x). Thus, ∫0 to 2 f(-x) dx + ∫0 to 2 f(x) dx = ∫0 to 2 -f(x) dx + ∫0 to 2 -f(x) dx =

∫0 to 2 -f(x) + f(x) dx = ∫0 to 2 0 dx = 0