How to prove: $f \in C^1([0, \pi])$ with $f(0)= f(\pi) = 0$, then $\int_0^{\pi} |f|^2 dx \le \int_0^{\pi} |f'|^2 dx$.
69428
Known proofs of Wirtinger's Inequality?
132448
Prove that: $\int_0^{n\pi}f(\cos^2(x))\,dx=n\int_0^{\pi}f(\cos^2(x))\,dx?$
40047
Is $\int_a^b f(x) dx = \int_{f(a)}^{f(b)} f^{-1}(x) dx$?
18261
Prove $\int_{a}^{b}{f(x)dx}>0$.
33247
Does $f(0)=0$ and $\left|f^\prime(x)\right|\leq\left|f(x)\right|$ imply $f(x)=0$?
33989
If $\int_0^1 f(x)x^n \ dx=0$ for every $n$, then $f=0$.
114734
Prove that $\frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2+\cdots+z^{2n}|^2~dz =2n$ where $\mathcal{C}$ is the unit circle
61162
Show that $\left(\int_0^1 f(x)\,dx\right)^2\leq2\int_0^1x\, f(x)^2\,dx $
13747
Proof of $\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}$
18905
$f(a)=f(b)=f_{+}'(a)=f_{-}'(b)=0$ and $|f''(x)|\le M$. Show that $|f(x)|\le \frac{M(b-a)^2}{16}$.
15262
Prove $\exists \xi, \eta \in (0,1)$, such that $f(\xi)+f'(\eta)=\xi+\eta$.
33081
If $\int_{\Bbb R^2}|f(x+iy)|\,dx\,dy<\infty$ then prove that $f\equiv 0$
69695
$f\geq 0$, continuous and $\int_a^b f=0$ implies $f=0$ everywhere on $[a,b]$
141852
Prove that $2\int_a^b \int_a^x f(x)f(y) \, dy \, dx = \left[ \int_a^b f(x) \, dx \right]^2$
70715
How do I prove $\int_0^\pi\frac{(\sin nx)^2}{(\sin x)^2}dx = n\pi$?
69008
$f'(x) = [f(x)]^{2}$. Prove $f(x) = 0 $
28949
Prove $f(x) = 0 $for all $x \in [0, \infty)$ when $|f'(x)| \leq |f(x)|$