query: If $\int_0^\pi{f(\theta)\cos\theta}=\int_0^\pi{f(\theta)\sin\theta}$, prove that $f$ has two zeroes in $(0,\pi)$
positive_passages.docid: 82567
positive_passages.text: Prove that $ f(x) $ has at least two real roots in $ (0,\pi) $
negative_passages.docid: 36366
negative_passages.text: Prove $\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}$
negative_passages.docid: 55738
negative_passages.text: Evaluating $\int _0^{\frac{\pi }{2}}\:\frac{\sqrt[3]{\sin^2\left(x\right)}}{\sqrt[3]{\sin^2\left(x\right)}+\sqrt[3]{\cos^2\left(x\right)}}dx$
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: 101030
negative_passages.text: Integration $I_n=\int _0^{\pi }\sin^{2n}\theta d\theta $
negative_passages.docid: 129042
negative_passages.text: Prove that $\lim_{a \to 0^{+}} \int_{0}^{a} \frac{1}{\sqrt{\cos(x)-\cos(a)}} \;dx=\frac{\pi}{\sqrt{2}}$
negative_passages.docid: 130967
negative_passages.text: How to show that $\int_{0}^{\infty}{\sin(x)\sin(2x)\sin(4x)\cdots\sin(2^kx)\over x^{k+1}}\mathrm dx=2^{0.5(k^2-k-2)}\pi?$
negative_passages.docid: 53307
negative_passages.text: how to strictly prove $\sin x<x$ for $0<x<\frac{\pi}{2}$
negative_passages.docid: 152336
negative_passages.text: Any hint to solve given integral $\int_0^{2{\pi}}{{d\theta}\over{a^2\cos^2\theta+b^2\sin^2\theta}}$?
negative_passages.docid: 96388
negative_passages.text: Prove $\sin(x)\tan(x) > x^2$ for $x \in ( \,0, \frac{\pi}{2}) \,$
negative_passages.docid: 31768
negative_passages.text: Roots of a functionwith condition $\int_0^\pi f(x) \sin x dx = \int_0^\pi f(x) \cos x dx =0.$
negative_passages.docid: 158279
negative_passages.text: $\int_0^{\pi/2}\ln(\sin(x))$?
negative_passages.docid: 136966
negative_passages.text: Riemann integral proof $\int^b_a f(x) \, dx>0$
negative_passages.docid: 151127
negative_passages.text: Confirm that $\int_{0}^{\infty}t^{-1}\sin t dt=\pi/2$
negative_passages.docid: 68421
negative_passages.text: Residue theorem with $\frac{1}{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) d\theta$.
negative_passages.docid: 70974
negative_passages.text: How can zero divided by $i$ be $2\pi$?
negative_passages.docid: 103598
negative_passages.text: Showing that $\int_0^1 \log(\sin \pi x)dx=-\log2$