query: Show that; $\displaystyle\int_0^\infty\frac{f(bx)-f(ax)}{x}dx=(l-f(0))\ln\frac{b}{a}$
positive_passages.docid: 717
positive_passages.text: Proof of Frullani's theorem
negative_passages.docid: 57686
negative_passages.text: Finding $ \int^1_0 \frac{\ln(1+x)}{x}dx$
negative_passages.docid: 129892
negative_passages.text: show $\int_0^{\infty} \frac{x\sin ax}{x^2+t^2}dx = \frac{\pi}{2}e^{-at}$
negative_passages.docid: 117246
negative_passages.text: Show that $\lim\limits_{a \to 0^+} \int^1_0 ax^{a−1}f(x)dx=f(0)$.
negative_passages.docid: 82961
negative_passages.text: Show that $\left| \frac{1}{x}-\frac{1}{z} \right| \le \left|\frac{1}{x} - \frac{1}{y} \right| + \left|\frac{1}{y} - \frac{1}{z} \right|$.
negative_passages.docid: 63251
negative_passages.text: Evaluation of $\int^1_0\frac{x^b-x^a}{\ln x} d x=\ln{{\frac{1+b}{1+a}}}$
negative_passages.docid: 18261
negative_passages.text: Prove $\int_{a}^{b}{f(x)dx}>0$.
negative_passages.docid: 123899
negative_passages.text: $\int_0^{+\infty}\frac{\sin(x)}{x^\alpha}dx=\frac{\Gamma(\alpha/2)\Gamma(1-\alpha/2)}{2\Gamma(\alpha)}$
negative_passages.docid: 106812
negative_passages.text: Find $\int_{0}^{\infty} \frac{\ln(x)}{1+x^2}dx$
negative_passages.docid: 85949
negative_passages.text: Integral involving logarithm: $\int_0^\infty \frac{ \ln x}{(x+a)(x+b)} dx$
negative_passages.docid: 131876
negative_passages.text: Show that $\int_{a}^{b}{x^{n}f(x)dx}=0$, then $f=0$
negative_passages.docid: 137060
negative_passages.text: Evaluating $\int_0^{\tfrac{\pi}{4}}\ln(\cos x-\sin x)\ln(\cos x) dx - \int_0^{\tfrac{\pi}{4}}\ln(\cos x+\sin x)\ln(\sin x)dx=\frac{G\ln 2}{2}$
negative_passages.docid: 67837
negative_passages.text: Evaluation of $\int_{0}^{1}\frac{x\ln (x)}{\sqrt{1-x^2}}dx$
negative_passages.docid: 35861
negative_passages.text: Evaluate $\int_0^\infty\frac{\ln x}{1+x^2}dx$
negative_passages.docid: 68241
negative_passages.text: Integral $\int_0^1 \frac{\ln(1-x)\ln(1+x^2)}{x}dx$
negative_passages.docid: 147977
negative_passages.text: Evaluate $\int_0^1\left(\frac{x-1}{x+1}\right)^n\frac{1}{\ln x} \,dx$
negative_passages.docid: 51284
negative_passages.text: $\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$ with residue calculus