query: prove $\int_a^\infty xf(x) \, dx \ge\int_a^\infty af(x) \, dx\,, \quad a>0, \, f\ge0$
positive_passages.docid: 149022
positive_passages.text: In Markov inequality proof, why is $\int_a^\infty xp(x) \, dx \ge \int_a^\infty ap(x) \, dx$
negative_passages.docid: 124026
negative_passages.text: Prove $\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$
negative_passages.docid: 30940
negative_passages.text: Prove the integral of $f$ is positive if $f ≥ 0$, $f$ continuous at $x_0$ and $f(x_0)>0$
negative_passages.docid: 54266
negative_passages.text: Prove that $\int_a^b \left( \int_c^d f(x,y)dy\right) dx=\int_c^d \left( \int_a^b f(x,y)dx\right) dy$
negative_passages.docid: 76689
negative_passages.text: How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?
negative_passages.docid: 40564
negative_passages.text: Prove that $u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw\rightarrow 0$ if $x\rightarrow \infty$
negative_passages.docid: 147108
negative_passages.text: Prove that $\int_0^\infty \frac{x\,\log x}{(1+x^2)^2} = 0$
negative_passages.docid: 115495
negative_passages.text: Prove $\int _0^\infty f^2 dx \leq \cdots $ for $f$ convex
negative_passages.docid: 145903
negative_passages.text: Prove that $\int_E |f_n-f|\to0 \iff \lim\limits_{n\to\infty}\int_E|f_n|=\int_E|f|.$
negative_passages.docid: 46844
negative_passages.text: Show that if $\int_0^x f(y)dy \sim Ax^\alpha$ then $f(x)\sim \alpha Ax^{\alpha -1}$
negative_passages.docid: 69124
negative_passages.text: Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$
negative_passages.docid: 6990
negative_passages.text: For $f$ Riemann integrable prove $\lim_{n\to\infty} \int_0^1x^nf(x)dx=0.$
negative_passages.docid: 84413
negative_passages.text: Why is $\int_{-\infty}^\infty |g(x)|\,dx = \int_0^\infty \mu(\{x : |g(x)| \ge t\})\,dt$ true?
negative_passages.docid: 20404
negative_passages.text: Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$
negative_passages.docid: 66404
negative_passages.text: Prove that:$\int_{a}^{b} f(x) dx = b \cdot f(b) - a \cdot f(a) - \int_{f(a)}^{f(b)} f^{-1}(x) dx$
negative_passages.docid: 79575
negative_passages.text: The sum of integrals of a function and its inverse: $\int_{0}^{a}f+\int_{0}^{f(a)}f^{-1}=af(a)$
negative_passages.docid: 113141
negative_passages.text: Prove that $\lim_{n \to \infty}\bigg[\int_0^1f(t)^n \text{dt}\bigg]^{1/n}=M$