query: Show that there is a $\xi\in(a,b)$ s.t. $\int_a^bf(x)g(x)\,dx=f(\xi)\int_a^b g(x)\,dx$
positive_passages.docid: 22516
positive_passages.text: Question about condtion of MVT for integral.
negative_passages.docid: 61133
negative_passages.text: Show that $\int|f(x)|dx=\int_0^\infty m(E_\alpha)d\alpha$
negative_passages.docid: 98344
negative_passages.text: Prove that $\int_0^1[f''(x)]^2dx\ge4.$
negative_passages.docid: 6450
negative_passages.text: Show $\mathbb{E}[f(X)g(X)] \geq \mathbb{E}[f(X)]\mathbb{E}[g(X)]$ for $f,g$ bounded, nondecreasing
negative_passages.docid: 97184
negative_passages.text: Show: $ f(a) = a,\ f(b) = b \implies \int_a^b \left[ f(x) + f^{-1}(x) \right] \, \mathrm{d}x = b^2 - a^2 $
negative_passages.docid: 43956
negative_passages.text: Differentiating $f(x)^{g(x)}$
negative_passages.docid: 93806
negative_passages.text: $\|f\|_{L^{p}}=\sup\left\{\int_{{\bf R}^{n}}|f(x)g(x)|~dx:\|g\|_{L^{q}}\le1\right\}$
negative_passages.docid: 5180
negative_passages.text: How to show $\alpha\int_a^cf(x)dx+\beta \int_c^b f(x)dx =0$ means $f(x)=0$ everywhere $?$
negative_passages.docid: 90369
negative_passages.text: Does $\int_a^bf(x) = 0$ imply $f(x) = 0$ for a Riemann Integral function?
negative_passages.docid: 73489
negative_passages.text: How is the Integral of $\int_a^bf(x)~dx=\int_a^bf(a+b-x)~dx$
negative_passages.docid: 34398
negative_passages.text: Prove $\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| dt$
negative_passages.docid: 117688
negative_passages.text: Prove that $\def\Aut{\operatorname{Aut}}\Aut(\mathbf{Z_{n}})\simeq \mathbf{Z_{n}^{*}}$
negative_passages.docid: 104394
negative_passages.text: What does $\int_{a}^{b} f(x)$ mean?
negative_passages.docid: 58832
negative_passages.text: How prove this $\int_{a}^{b}[f''(x)]^2dx\ge\dfrac{4}{b-a}$
negative_passages.docid: 70512
negative_passages.text: Prove that $\int f\ d\lambda = \int_{a}^{b} f(x)\ dx,$ for any $f \in \mathcal R[a,b].$
negative_passages.docid: 40047
negative_passages.text: Is $\int_a^b f(x) dx = \int_{f(a)}^{f(b)} f^{-1}(x) dx$?
negative_passages.docid: 62270
negative_passages.text: Given two increasing continuous functions $f,g$ prove that $(b-a) \int^b_a f(x)g(x) dx > \int^b_a f(x) dx \int^b_a g(x) dx$