New posts in Integral Inequality

Vector norm of integral inequality

Integral inequality - École Polytechnique International Entrance Exam

Prove that $\left|30240\int_{0}^{1}x(1-x)f(x)f'(x)dx\right|\le1$.

Prove that $1<\int_{0}^{\frac{\pi}{2}}\sqrt{\sin x}dx<\sqrt{\frac{\pi}{2}}$using integration.

Show that $|a|\leq \int_{-1}^{1}\,|ax+b|\,\text{d}x$

Show $\int_0^t (t-x)P_n(x)\,dx\leq \frac{t^2}{2}\int_0^1 P_n(x)\,\mathrm dx $ where $P_n(x)=(x(1-x))^{n}$

Given $\int_{\frac13}^{\frac23}f(x)dx=0$, how to prove $4860(\int_0^1f(x)dx)^2\le 11\int_0^1|f''(x)|^2dx$?

Finding the maximum value of $\int_0^1 f^3(x)dx$

Prove that $\int_0^1 \big(1-x^2\big) \big(f'(x)\big)^2\,dx \ge 24 \left(\int_0^1 xf(x)\,dx\right)^{\!2}$

A problem from the Shortlist of the Romanian Mathematics Olympiad