query: Proving $\int_0^\sqrt{2\pi}\sin x^2\,\mathrm dx>0$
positive_passages.docid: 156881
positive_passages.text: How prove this inequality $I=\int_{0}^{\sqrt{2\pi}}\sin{x^2}dx>0$
negative_passages.docid: 41191
negative_passages.text: How to prove this $\pi$ formula?
negative_passages.docid: 4287
negative_passages.text: Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$
negative_passages.docid: 50509
negative_passages.text: Prove that $\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}}~dx=0$ (without trigonometric substitution)
negative_passages.docid: 36366
negative_passages.text: Prove $\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}$
negative_passages.docid: 105622
negative_passages.text: How to show $ \sin x \geq \frac{2x}{\pi}, x \in [0, \frac{\pi}{2}]$?
negative_passages.docid: 131120
negative_passages.text: Proving an integral equation.
negative_passages.docid: 109295
negative_passages.text: Proving $1-\exp(-4x^2/ \pi) \ge \text{erf}(x)^2$
negative_passages.docid: 90313
negative_passages.text: Integrate $\int_0^\pi{{x\sin x}\over{1+\cos^2x}}dx$.
negative_passages.docid: 117264
negative_passages.text: Prove that $\int_0^{\infty} \frac{\sin x}{x^p}\, dx$ converges for $0<p<2$
negative_passages.docid: 62173
negative_passages.text: How do I show that $\int_0^\infty \frac{\sin(ax) \sin(bx)}{x^{2}} \, \mathrm dx = \pi \min(a,b)/2$
negative_passages.docid: 51837
negative_passages.text: Show that $\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$
negative_passages.docid: 42805
negative_passages.text: Show that $\int\limits_0^1 \left(x^{x}\right)^{\left(x^{x}\right)^{\left(x^{x}\right)^{\left(x^{x}\right)^{⋰}}}}\ \mathrm{d}x=\frac{\pi^2}{12}$.
negative_passages.docid: 146180
negative_passages.text: Prove that $x\sqrt{1-x^2} \leq \sin x \leq x$
negative_passages.docid: 154264
negative_passages.text: $f \in C[-1,1]$, Prove ${\lim_{h \to 0^+}}{\int_{-1}^1 \frac{h}{h^2+x^2}f(x)\,dx} = \pi f(0)$
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: 60559
negative_passages.text: How prove this $x^3<\sin^2{x}\tan{x},x\in\left(0,\dfrac{\pi}{2}\right)$