1

Let $f:[−1, 1] \to \mathbb{R}$ is continuous. Suppose we have $$ \int_{−1}^1 \sin^n(x)f(x) dx=0 \quad \forall n \in \mathbb{Z}^+. $$ Show that $f(x) =0$ on $[-1,1]$.

I tried using $u = \sin(x)$, but it didn't really get me anywhere yet. Please, some insight would be gracious.

gt6989b
  • 54,422
  • 1
    If its true for all n, then in particular its true for even n, which should be easy to prove since $sin^{2n}$ is non-negative (equal to 0 at only one point too, which is important), then since f must be identically 0 for even n, and 0 also works for odd n, then f must be identically 0 – Displayname Feb 28 '19 at 23:00
  • 3
    Tip: use approach0 to find out if the question has been asked already. In this case it seems a class mate asked the same question 5 hours ago. – ViktorStein Feb 28 '19 at 23:35

0 Answers0