# Evaluating $\displaystyle\lim_{n \to +\infty}\int _{-1}^1 f(t)\cos^2(nt) \mathrm dt$

Let $f:[0,1]\rightarrow \mathbb R$ be continuous. Assume that $\displaystyle\int_{-1}^{1}f(t)\mathrm dt=1$. Evaluate

$$\lim_{n \to +\infty}\int _{-1}^1 f(t)\cos^2(nt) \mathrm dt$$

How to evaluate this? Integration by parts is not yielding anything.

• First of all, what do you imagine the answer will be? – Simon S Dec 8 '14 at 13:18
• See this. – David Mitra Dec 8 '14 at 13:19
• I have no idea how to imagine an answer of a mathematical problem@SimonS – Learnmore Dec 8 '14 at 13:23
• @learning maths, do you want me to detail more? Both links have complete proofs of the lemma, are they clear to you? It would be best that you precise what degree of knowledge you've reached, so that answers can be fit to your situation – mvggz Dec 8 '14 at 13:26
• thanks @DavidMitra for the link – Learnmore Dec 8 '14 at 13:29

1) $\cos^2(u) = \frac{1-cos(2u)}{2}$
2) Riemann's lemma, that is being proven here (you don't integrate on the same interval but the proof is exactly the same): lim$_{n\rightarrow \infty}\int _{-\pi}^\pi f(t)\cos nt\,dt$