query: How can I show that $\int_{0}^1\int_{0}^1 \frac{1}{1-xy}dxdy=\frac{\pi^2}{6}$ through substitution?
positive_passages.docid: 29886
positive_passages.text: How to evaluate $\int_0^1\int_0^1 \frac{1}{1-xy} \, dy \, dx$ to prove $\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$.
negative_passages.docid: 34762
negative_passages.text: Why isn't $\int \frac{1}{x}~dx = \frac{x^0}{0}$?
negative_passages.docid: 36701
negative_passages.text: $\displaystyle\int_{0}^{2\pi}\frac{1}{a+ \cos\theta}\,d\theta$
negative_passages.docid: 112621
negative_passages.text: Finding $\int_{0}^{\infty} \frac{e^{3x}-e^x}{x(e^{3x}+1)(e^x+1)} dx$
negative_passages.docid: 151344
negative_passages.text: Finding $\int_{0}^{1}\int_{0}^{1}\int_{0}^{1} \left\{\frac{x}{y}\right\} \left\{\frac{y}{z}\right\}\left\{\frac{z}{x}\right\} dx\space dy\space dz $
negative_passages.docid: 142619
negative_passages.text: Evaluating the complex integral $\int_{|z|=2}^{}\frac{1}{z^2+1}dz$
negative_passages.docid: 124677
negative_passages.text: Substitution needed for calculating an integral
negative_passages.docid: 16775
negative_passages.text: Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$
negative_passages.docid: 31063
negative_passages.text: Evaluating $\int_0^2(\tan^{-1}(\pi x)-\tan^{-1} x)\,\mathrm{d}x$
negative_passages.docid: 112551
negative_passages.text: Demystify integration of $\int \frac{1}{x} \mathrm dx$
negative_passages.docid: 1626
negative_passages.text: How do we prove $\int_I\int_x^1\frac{1}{t}f(t)\text{ dt}\text{ dx}=\int_If(x)\text{ dx}$
negative_passages.docid: 154185
negative_passages.text: Evaluate $\iiint_{[0,1]^3}\frac{dx\,dy\,dz}{(1+x^2+y^2+z^2)^2}$
negative_passages.docid: 70715
negative_passages.text: How do I prove $\int_0^\pi\frac{(\sin nx)^2}{(\sin x)^2}dx = n\pi$?
negative_passages.docid: 84034
negative_passages.text: Evaluate $\int_{-\infty}^{\infty} \chi_{[0,1]}(x-y) \chi_{[0,1]}(y) \, \mathrm{d}y$
negative_passages.docid: 103598
negative_passages.text: Showing that $\int_0^1 \log(\sin \pi x)dx=-\log2$
negative_passages.docid: 113066
negative_passages.text: Calculate $\int_0^1f(x)dx$
negative_passages.docid: 1744
negative_passages.text: Quadruple Integral $\int\limits_0^1\!\!\int\limits_0^1\!\!\int\limits_0^1\!\!\int\limits_0^1\frac1{1-xyzw}\,dw\,dz\,dy\,dx$