Prove $\int_{0}^{1}|f(x)|^2\,dx\le\int_{0}^{1}|f{'}(x)|^2\,dx$ with $f(0)=0$
4656
Prove that there is a $\delta$ such that $\int_{0}^{1} (f(x))^2dx\leq \delta$$\int_{0}^{1} (f'(x))^2dx$ for all $f$ with these conditions
34398
Prove $\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| dt$
97643
Prove that:$f(f(x)) = x^2 \implies \int_{0}^{1}{(f(x))^2dx} \geq \frac{3}{13}$
128790
Prove an integral inequality: $ \left(\int|f|^2dx\right)^2\le 4\left(\int|xf(x)|^2dx\right)\left(\int|f'|^2dx\right) $
35325
Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.
58832
How prove this $\int_{a}^{b}[f''(x)]^2dx\ge\dfrac{4}{b-a}$
57695
Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.
18261
Prove $\int_{a}^{b}{f(x)dx}>0$.
13202
Prove that there exists at least one $x_0\in\mathbb{R}$, such that $f(x_0)+f''(x_0)=0$
5180
How to show $\alpha\int_a^cf(x)dx+\beta \int_c^b f(x)dx =0$ means $f(x)=0$ everywhere $?$
58647
Show that $f$ is integrable and $\int_{I} f=0$
136966
Riemann integral proof $\int^b_a f(x) \, dx>0$
46310
Showing that $ \int_{0}^{1} \frac{x-1}{\ln(x)} \mathrm dx=\ln2 $
91775
Show that $\int_0^1 f^2(x)dx\geq 3$, if $f:[0,1]\to \mathbb{R}$ is an integrable function s.t. $\int_0^1 f(x)dx=\int_0^1 xf(x)dx=1$
1626
How do we prove $\int_I\int_x^1\frac{1}{t}f(t)\text{ dt}\text{ dx}=\int_If(x)\text{ dx}$
66504
Proof that $f$ is Lebesgue-integrable and calculate its integral
97253
How prove $e^x|f(x)|\le 2$ if $f(x)=\int_{x}^{x+1}\sin{(e^t)}dt$