query: Value of $5050\cdot\int^1_0(1-x^{50})^{100} dx/\int^1_0(1-x^{50})^{101} dx$
positive_passages.docid: 143368
positive_passages.text: Find the ratio of $\frac{\int_{0}^{1} \left(1-x^{50}\right)^{100} dx}{\int_{0}^{1} \left(1-x^{50}\right)^{101} dx}$
negative_passages.docid: 84990
negative_passages.text: Value of $x=\sum\limits_{n=1}^\infty {\frac{1}{10^n-1}} $ and location of second digit $1$ of $x$
negative_passages.docid: 70790
negative_passages.text: Evaluation of $\lim_{q \to 0} (\int_0^1 |f|^q)^{1/q} $ for $f \in L^1([0,1])$.
negative_passages.docid: 139121
negative_passages.text: Evaluating $\int _0 ^1 x^k(\ln x)^m dx$ for integer $k$ and $m$
negative_passages.docid: 72620
negative_passages.text: Finding the value of $1.1^{82}$ using $(1+x)^{82}$ to a certain accuracy
negative_passages.docid: 86843
negative_passages.text: Find the value of : $\lim_{n\to\infty}[(n+1)\int_{0}^{1}x^{n}\ln(1+x)dx]$.
negative_passages.docid: 148613
negative_passages.text: Integrate $\int_{0}^{1}{x^{-x}(1-x)^{x-1}\sin{\pi x}dx}$
negative_passages.docid: 121071
negative_passages.text: Calculation of $\int^{\pi/2}_{0}\cos^{n}(x)\cos (nx)\,dx$, where $n\in \mathbb{N}$
negative_passages.docid: 15436
negative_passages.text: Find p and q s.t. $({p-1})^{q-1}+({q-1})^{p-1} \equiv 101 \pmod {pq}$
negative_passages.docid: 69978
negative_passages.text: Evaluating $\int_1^\infty \frac{1}{x(x^2+1)}\ dx$
negative_passages.docid: 113066
negative_passages.text: Calculate $\int_0^1f(x)dx$
negative_passages.docid: 125512
negative_passages.text: Doubt regarding divisibility of the expression: $1^{101}+2^{101} \cdot \cdot \cdot +2016^{101}$
negative_passages.docid: 156956
negative_passages.text: Give the remainder of $x^{100}$ divided by $(x-2)(x-1)$.
negative_passages.docid: 57686
negative_passages.text: Finding $ \int^1_0 \frac{\ln(1+x)}{x}dx$
negative_passages.docid: 64216
negative_passages.text: Is $0.1010010001000010000010000001 \ldots$ transcendental?
negative_passages.docid: 99918
negative_passages.text: The $100$th derivative of $(x^2 + 1)/(x^3 - x)$
negative_passages.docid: 100711
negative_passages.text: $\int_{-\infty}^{\infty}e^{-zt^2 + 2wt} \ dt = e^{w^2 / z} \sqrt{\frac{\pi}{z}}$ for $z,w \in \mathbb{C}, Re(z) > 0$