query: How to prove $ \mathrm{e}^x\left|\int_x^{x+1}\sin\mathrm e^t \,\mathrm d t\right|\leqslant 2$?
positive_passages.docid: 28053
positive_passages.text: Integral of $\sin(e^t)dt$
negative_passages.docid: 15324
negative_passages.text: A sine integral $\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$
negative_passages.docid: 10036
negative_passages.text: Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$
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: 68821
negative_passages.text: Prove $\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$
negative_passages.docid: 127291
negative_passages.text: Show that $\left( \frac{q}{p} \right) \equiv q^{(p-1) / 2} \mod p$, where $\left( \frac{q}{p} \right)$ is the Legendre Symbol
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: 8829
negative_passages.text: How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$
negative_passages.docid: 105145
negative_passages.text: Show that $\mathbb{E}\left(\bar{X}_{n}\mid X_{(1)},X_{(n)}\right) = \frac{X_{(1)}+X_{(n)}}{2}$
negative_passages.docid: 121548
negative_passages.text: proof of an inequality $ \left( 1+ \frac{x}{p} \right) ^{p}$
negative_passages.docid: 140459
negative_passages.text: Why is $\left(e^{2\pi i}\right)^i \neq e^{-2 \pi}$?
negative_passages.docid: 97558
negative_passages.text: Prove that $\left| f'(x)\right| \leq \sqrt{2AC}$ using integration
negative_passages.docid: 21088
negative_passages.text: Prove $\int\limits_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$
negative_passages.docid: 35319
negative_passages.text: I'm looking for several ways to prove that $\int_{0}^{\infty }\sin(x)x^mdx=\cos(\frac{\pi m}{2})\Gamma (m+1)$
negative_passages.docid: 24683
negative_passages.text: For i.i.d. $\{X_n\}$ proving $\frac1n\max\limits_{1\leqslant k\leqslant n}|X_k|\xrightarrow{\mathrm{a.s.}}0\Leftrightarrow E(|X_1|)<+∞$
negative_passages.docid: 156881
negative_passages.text: How prove this inequality $I=\int_{0}^{\sqrt{2\pi}}\sin{x^2}dx>0$
negative_passages.docid: 27347
negative_passages.text: Verify that $\left|\int_{\gamma} \exp(iz^2)dz\right| \leq \frac{\pi\big(1-\exp(-r^2)\big)}{4r}$ where $\gamma(t)=re^{it}$, for $0\leq t \leq \pi/4$.