# Tagged Questions

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### Evaluate $\int_{1}^{\infty} \frac{\ln{(2x-1)}}{x^2}$

$$\int_{1}^{\infty} \frac{\ln{(2x-1)}}{x^2} dx$$ My approach is to calc $$\int_{1}^{X} \frac{\ln{(2x-1)}}{x^2} dx$$ and then take the limit for the answer when $X \rightarrow \infty$ However, I must ...
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### A closed form for $\int_{0}^{\pi/2}\frac{\ln\cos x}{x}\mathrm{d}x$?

The following integrals are classic, initiated by L. Euler. \begin{align} \displaystyle \int_{0}^{\pi/2} x^3 \ln\cos x\:\mathrm{d}x & = -\frac{\pi^4}{64} \ln 2-\frac{3\pi^2}{16} ...
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### Evaluating $\int_0^\pi\arctan\left(\frac{\ln\sin x}{x}\right)\mathrm{d}x$

I found the following integral as a by product of another one. It has a nice closed form. $$\int_{0}^{\pi} \arctan\left(\ln\left(\sin x \right) \over x\right)\,{\rm d}x$$ Mathematica and ...
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### Logarithmic integral and natural numbers.

Prove these two relations: $$\text{li}(k+1)+k-\log (k)-\gamma = \int_0^k \left(\int_1^2 \frac{(s+1)^{n-1}+s-1}{s} \, dn\right) \, ds$$ $$n = \lim_{s\to 0} \, \frac{(s+1)^{n-1}+s-1}{s}$$ ...
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### calculate $\int_{0}^{\pi} \int_{0}^{x}\log(\sin(x-y))dydx$

I was asked to find the integral $\iint_A \log(\sin(x-y))dxdy$ where $A$ is the triangle $y=0, x=\pi, y=x$ in the first quadrant. I was given a hint: evaluate $\int_{0}^{\pi}\log(\sin(t))dt$ using ...
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### What is ${\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$?

This is a new integral that I propose to evaluate in closed form: $${\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$$ where $\log (z)$ denotes the principal value of ...
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### A logarithm integral

Calculate the integral \begin{align} \int_{0}^{1} \frac{ \ln(\sqrt{x} - \sqrt{1-x}) }{ \sqrt{x} } \ dx \end{align} and show the value is negative.
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The integral \begin{align} I_{4} = \int_{0}^{1} \ln(1-x) \ \ln^{2}\left( \ln\left(\frac{1}{x}\right) \right) \ \frac{dx}{x} \end{align} can be expressed as \begin{align} I_{4} = \zeta^{''}(2) - ...
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### Hints on calculating the integral $\int_0^1\frac{x^{19}-1}{\ln x}\,dx$

I would be happy to get some hints on the following integral: $$\int_0^1\frac{x^{19}-1}{\ln x}\,dx$$
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So I'm to find the definite integral of a function which I'm to convert into partial fractions. $$\int_0^1 \frac{2}{2x^2+3x+1}\,dx$$ Converting to partial fractions I get... $\frac{A}{2x+1} + ... 2answers 244 views ### Proof$e^x = \exp(x)$? Define $$\ln (x) = \int^{x}_{1}\frac{1}{t}$$ Assume I have proven that$\ln x$is one-to-one and therefore has an inverse$\exp (x)$. Define$e$as:$\ln e = 1$Now, if you have no other notion ... 2answers 71 views ### Scale-invariance of Simpson's rule approximations to log If I was trapped on a desert island and needed to compute$\log(2)$, the natural logaritm of$2$, one thing I could do is use the equality $$\log(2) = \int_1^2 \frac{1}{x} \ dx$$ and approximate the ... 1answer 50 views ### Solve$f(x) = x^n$for$n$given the value of the definite integral of$f(x)$between some values$a$and$b$I have a curve$f(x) = x^n$, where$n$is variable. The curve will be plotted from the$f(a)$to$f(b)$, where$a$and$b$are positive integers, and$b > a$. I want to find$n$given the desired ... 4answers 1k views ### Integral$\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}dx$Is there a closed form for the integral $$\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}dx.$$ I do not have a strong reason to be sure it exists, but I would be ... 2answers 1k views ### Closed form for$\int_0^1\sqrt{\frac{2-x}{(1-x)\,x}}\,\log\left(\frac{(2-x)\,x}{1-x}\right)dx$This is somewhat similar to my previous question: Closed form for$\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$Is it possible to find a closed form ... 2answers 381 views ### Closed form for$\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$I need to evaluate this integral: $$Q=\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx.$$ I tried it in Mathematica, but it was not able to find a closed ... 2answers 310 views ### Integral$\int_0^1\frac{\ln x}{x^2+1}\cdot\ln\left(\frac{3\,x^2+1}{x^2+3}\right)dx$I need to evaluate the following integral: $$\int_0^1\frac{\ln x}{x^2+1}\cdot\ln\left(\frac{3\,x^2+1}{x^2+3}\right)dx.$$ Could you suggest how to find a closed form for it? I am not sure if there is ... 3answers 529 views ### Integral$\int_0^1\frac{\ln x}{\left(1+x\right)\left(1+x^{-\left(2+\sqrt3\right)}\right)}dx$There is a curious known integral: $$\int_0^1\frac{\ln\left(1+x^{2+\sqrt{3\vphantom{\large3}}}\right)}{1+x}dx=\frac{\pi^2}{12}\left(1-\sqrt{3\vphantom{\large3}}\right)+\ln ... 1answer 727 views ### Prove \int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}dx=\frac{\pi^2}8-\frac12 How can I prove the following identity?$$\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}dx=\frac{\pi^2}8-\frac12$$2answers 743 views ### Integral \int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx Please help me to evaluate this integral:$$\large\int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx$$3answers 480 views ### A closed form for \int_0^\infty\frac{\ln(x+4)}{\sqrt{x\,(x+3)\,(x+4)}}dx I need to a evaluate the following integral$$I=\int_0^\infty\frac{\ln(x+4)}{\sqrt{x\,(x+3)\,(x+4)}}dx.$$Both Mathematica and Maple failed to evaluate it in a closed form, and lookups of the ... 1answer 262 views ### Integral \int_0^\infty\frac{1}{\sqrt[3]{x}}\left(1+\log\frac{1+e^{x-1}}{1+e^x}\right)dx Is it possible to evaluate this integral in a closed form?$$\int_0^\infty\frac{1}{\sqrt[3]{x}}\left(1+\log\frac{1+e^{x-1}}{1+e^x}\right)dx$$1answer 141 views ### An integral involving the inverse of f(x)=\log x-\log\cos x+x\tan x Let the function f:\left(0,\,\displaystyle\frac\pi2\right)\to\mathbb{R} be defined as$$f(x)=\log x-\log\cos x+x\tan x.$$Let its inverse be denoted as ... 2answers 96 views ### Evaluating \int _{-1}^{e} \frac{1}{x}dx Here very easily by the Fundamental Theorem of Calculus$$\int _{-1}^{e} \frac{1}{x}dx=\ln(e)-\ln(-1)$$From Euler's identity e^{i \pi}=-1 we can easily deduce that \ln(-1)=i \pi. Thus the ... 2answers 719 views ### Are there other cases similar to Herglotz's integral \int_0^1\frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\ \mathrm dt? This post of Boris Bukh mentions amazing Gustav Herglotz's integral$$\int_0^1\frac{\ln\left(1+t^{\,4\,+\,\sqrt{\vphantom{\large A}\,15\,}\,}\right)}{1+t}\ \mathrm ... 9answers 305 views ### Finding the definite integral$\int_0^1 \log x\,\mathrm dx$$$\int_{0}^1 \log x \,\mathrm dx$$ How to solve this? I am having problems with the limit$0$to$1$. Because$\log 0$is undefined. 0answers 59 views ### Expressing$\int _0^1da\int _0^{1-a}\ln ((a-1+b)^2-4 y a b )+\int_0^1 da\int _0^{1-a}\frac{(a-1+b)^2}{(a-1+b)^2-4 y a b}db$in terms of dilogarithms I came across these integrals and I'm trying to rewrite them in terms of Dilogarithms:$\mathrm{Li}_2(z):=-\int_0^z \frac{\mathrm{d}s}{s}\log(1-s)$. Can anyone suggest how to contunue? If there is a ... 1answer 235 views ### What is a closed form of$\int_0^1\ln(-\ln x)\ \text{li}\ x\ dx$Let$\operatorname{li} x$denote the logarithmic integral: $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Is it possible to find a closed form of the following integral? $$\int_0^1\ln(-\ln x) ... 1answer 616 views ### Closed form for \int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx I encountered this integral in my calculations:$$\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm ... 2answers 1k views ### Closed form for$\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx\$
Please help me to find a closed form for the following integral: $$\int_0^1\log\left(\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\right)\,{\mathrm d}x.$$ I was told it could be calculated in a ...