query: Prove $2\int_a^b\left[f(x)\int_x^bf(y)\,dy\right]dx=\left[\int_a^bf(x)\,dx\right]^2$
positive_passages.docid: 141852
positive_passages.text: Prove that $2\int_a^b \int_a^x f(x)f(y) \, dy \, dx = \left[ \int_a^b f(x) \, dx \right]^2$
negative_passages.docid: 34017
negative_passages.text: Show that $n^{\pi\left(2n\right)-\pi\left(n\right)}2$
negative_passages.docid: 78174
negative_passages.text: How prove this inequality $\left(\int_{0}^{1}f(x)dx\right)^2\le\frac{1}{12}\int_{0}^{1}|f'(x)|^2dx$
negative_passages.docid: 138069
negative_passages.text: Proof that $\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $
negative_passages.docid: 137664
negative_passages.text: What exactly is $F$ in $W = \int_{a}^{b} F dx$?
negative_passages.docid: 136966
negative_passages.text: Riemann integral proof $\int^b_a f(x) \, dx>0$
negative_passages.docid: 58832
negative_passages.text: How prove this $\int_{a}^{b}[f''(x)]^2dx\ge\dfrac{4}{b-a}$
negative_passages.docid: 18126
negative_passages.text: If $f \colon [a,b] \rightarrow \mathbb{R}$, then show that there exists $c \in [a,b]$ such that $\int_a^c f = \int_c^b f$
negative_passages.docid: 25648
negative_passages.text: Prove $(a_1+b_1)^{1/n}\cdots(a_n+b_n)^{1/n}\ge \left(a_1\cdots a_n\right)^{1/n}+\left(b_1\cdots b_n\right)^{1/n}$
negative_passages.docid: 64277
negative_passages.text: How to prove ${{a}^{a}}{{b}^{b}}\ge {{\left(\frac{a+b}{2}\right)}^{a+b}}$ ?thanks.
negative_passages.docid: 98344
negative_passages.text: Prove that $\int_0^1[f''(x)]^2dx\ge4.$
negative_passages.docid: 133990
negative_passages.text: How prove this $\int_{a}^{b}f(x)dx=\frac{1}{2}(b-a)[f(a)+f(b)]-\frac{1}{12}(b-a)^3f''(\xi)$
negative_passages.docid: 18261
negative_passages.text: Prove $\int_{a}^{b}{f(x)dx}>0$.
negative_passages.docid: 110368
negative_passages.text: Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$
negative_passages.docid: 97558
negative_passages.text: Prove that $\left| f'(x)\right| \leq \sqrt{2AC}$ using integration
negative_passages.docid: 128790
negative_passages.text: Prove an integral inequality: $ \left(\int|f|^2dx\right)^2\le 4\left(\int|xf(x)|^2dx\right)\left(\int|f'|^2dx\right) $
negative_passages.docid: 40047
negative_passages.text: Is $\int_a^b f(x) dx = \int_{f(a)}^{f(b)} f^{-1}(x) dx$?