query: Prove that $\int_a^bf(x)dx+ \int_{f(a)}^{f(b)} f^{-1}(y)dy=bf(b)-af(a)$
positive_passages.docid: 138074
positive_passages.text: Integrating the inverse of a function.
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: 97643
negative_passages.text: Prove that:$f(f(x)) = x^2 \implies \int_{0}^{1}{(f(x))^2dx} \geq \frac{3}{13}$
negative_passages.docid: 105569
negative_passages.text: If $f$ is integrable in $[a-1,b+1]$, then prove that $\lim \limits_{h\to 0} \ \int_{a}^{b}|f(x+h)-f(x)| ~ dx=0$
negative_passages.docid: 62270
negative_passages.text: Given two increasing continuous functions $f,g$ prove that $(b-a) \int^b_a f(x)g(x) dx > \int^b_a f(x) dx \int^b_a g(x) dx$
negative_passages.docid: 1588
negative_passages.text: How to prove $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$
negative_passages.docid: 78834
negative_passages.text: if f(x) if differentiable and continuous, prove $\frac{af(a)-bf(b)}{a-b} = f(c) + cf'(c) $
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: 60348
negative_passages.text: How do I prove $F(a)=F(a^2)?$
negative_passages.docid: 55364
negative_passages.text: Show $au_x+bu_y=f(x,y)$ gives $u(x,y)=(a^2+b^2)^{-\frac{1}{2}}\int_{L}fds +g(bx-ay)$ if $a\neq 0$.
negative_passages.docid: 45602
negative_passages.text: Show that is $\int_A f$ exists, then so does $\int_A |f|$
negative_passages.docid: 69656
negative_passages.text: Prove or disprove that $\int_a^b |f(x)|\ \mathrm{d}x\geq \big | \int_a^b f(x)\ \mathrm{d}x\big| $
negative_passages.docid: 16109
negative_passages.text: Prove that $ \int _a^b\:f\left(x\right)\:\:dx\:=\:\int _a^b\:f\left(a+b-x\right)\:\:dx $
negative_passages.docid: 94869
negative_passages.text: Show that $ \int_{k}^{2} \frac{1}{f(y)} =1$
negative_passages.docid: 27705
negative_passages.text: Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$
negative_passages.docid: 61133
negative_passages.text: Show that $\int|f(x)|dx=\int_0^\infty m(E_\alpha)d\alpha$
negative_passages.docid: 131120
negative_passages.text: Proving an integral equation.