Question

Let f be an odd function, that is f(-x)=-f(x) . We know that integral 0^5 f(x)=3. Then using properties of definite integrals we conclude that

integral -5^5 f(x)= -3
integral -5^5 f(x)= 0
integral -5^5 f(x)= 2
integral -5^5 f(x)= 3

I'm confused on how to set it up from the integral range they are giving how would this affect the answer?

Answer 1

We're given \[\Large \int_{0}^5f(x)dx = 3\]

Answer 2

We want to find \[\Large \int_{-5}^5f(x)dx\]

Answer 3

So

\[\Large \int_{-5}^5f(x)dx\]

\[\Large \int_{-5}^0f(x)dx+\int_{0}^5f(x)dx\]

\[\Large \int_{0}^5f(-x)dx+\int_{0}^5f(x)dx\]

\[\Large \int_{0}^5-f(x)dx+\int_{0}^5f(x)dx\]

\[\Large -\int_{0}^5f(x)dx+\int_{0}^5f(x)dx\]

\[\Large -3+3\]

\[\Large 0\]

Which means that

\[\Large \int_{-5}^5f(x)dx = 0\]

Answer 4

Oh ok, I wasn't sure how to break it up. Now that you set it up it makes complete sense I can't believe I overlooked that. Thanks!!

Answer 5

you're welcome