# Tagged Questions

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### Is this double limit for logarithms true?

Mathematica knows that: $$\gamma = \lim_{n\to \infty } \, \lim_{s\to 0} \, \left(\int \frac{(s+1)^{-\exp (n)-1}+s-1}{s} \, ds+\frac{(s+1)^{-n-1}+s-1}{s}\right)$$ Where $\gamma$ is Euler Gamma ...
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### $\int_0^1\frac{1-t}{(t-2)\ln t}\,dt$ integral

I have two related questions. The first is: Is there a closed form expression for: $$\int_0^1\frac{1-t}{(t-2)\ln t}\,dt\approx0.507834$$ I know that there are some very superb integrators on this ...
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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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I have to prove the following relations: $\sum_{x=a}^{b-1}\frac{1}{x}\geq\log b - \log a$ $\sum_{x=a+1}^{b}\frac{1}{x}\leq\log b - \log a$ I tried to use the relation that $\int_a^b \frac{1}{x} ... 1answer 131 views ### What is${\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 ... 2answers 71 views ### Commutation between Logarithm and Gaussian Integral. I'm calculating a partition function (physics) and I arrive to the following expression: $$\log \int_{-\infty}^{\infty} \frac{du}{\sqrt{2\pi}} e^{-u^2/2} e^{-nq/2}[2\cosh(\sqrt{q}\,u+m)]^n \qquad(1)$$ ... 0answers 57 views ### Logarithm and “basic” functions. To express the antiderivatives of$\frac{1}{x}$, we cannot apply the formula$\int x^n dx=\frac{x^{n+1}}{n+1}+C$and we need to introduce a new function, the logarithm. But how can we prove that ... 1answer 70 views ###$\iiint \frac{1}{x^2+y^2+(z-2)^2}dA$where$A=\{x^2+y^2+z^2 \leq 1\}$check my answer! I would like someone to review my solution please, the original question is to calculate$\iiint \frac{1}{x^2+y^2+(z-2)^2}dA$where$A=\{x^2+y^2+z^2 \leq 1\}$What I did: First I changed variables ... 1answer 94 views ### Unable to comprehend a connection between two equations I was reading this paper and got stuck at the transition from Equation (13) to Equation (14) (p. 16/17). We got a function of the form:$y(t)=k(t)^{\alpha}h(t)^{\beta}$We know it grows from zero ... 3answers 62 views ### Two different solutions of the same integral Considering $$\int\frac{\ln(x+1)}{2(x+1)}dx$$ I first solved it seeing it similar to the derivative of$\ln^2(x+1)$so multiplying by$\frac22$the solution is ... 4answers 363 views ### 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: $$\int_0^1\frac{x^{19}-1}{\ln x}\,dx$$ 3answers 64 views ### Why does$\int^{ab}_{a} \frac{1}{x} dx = \int^{b}_{1} \frac{1}{t} dt$? I can't understand how the integral having limits from$a$to$ab$in Step 1 is equivalent to the integral having limits from$1$to$b$. I'm a beginner here. Please explain in detail. ... 1answer 41 views ### Evaluate$\lim\limits_{x\to\infty}\frac{1}{\sqrt{x}}\int_1^x\ln(1+\frac{1}{\sqrt{t}})dt\lim\limits_{x\to\infty}\frac{1}{\sqrt{x}}\displaystyle\int_1^x\ln(1+\frac{1}{\sqrt{t}})dt=?$If the limit exists with l'Hopital i get ... 2answers 77 views ### Integral of$\frac{1}{x^2+1}using complex partial fractions. Is there any way to evaluate the following integral via a complex partial fraction decomposition? $$\int \dfrac{1}{x^2 + 1} \text{ d}x$$ So far I have: \begin{aligned} \int \dfrac{1}{x^2 + 1} ... 2answers 47 views ### Integral of e^x ln(e^{2x} - 4) Find the integral from ln4 to ln6 ofe^x \ln(e^{2x} - 4)$$I factored$$\ln(e^{2x} - 4)$$to get$$\ln((e^{x} - 2)(e^{x} + 2))$$Then I separated this to get:$$e^x\ln(e^{x} - 2) + e^x\ln(e^{x} + ... 0answers 53 views ### Integral of Difference of Logs I get the expansion ofh$to be $$h(z) = {1 \over z } \sum_{r=1}^{\infty}{1 \over r}{(-{\alpha \over z}})^r$$ $$\Rightarrow h(z) = \sum_{r=-2}^{-\infty}{{(-\alpha)^{r+1} \over -(r+1)} z^{r}}$$ ... 2answers 117 views ### How to find$\int_2^x t/(\log t)^2 \,dt$$$\int_2^x \frac{t}{(\log t)^2} \,dt,$$ I want to write this integral with$Li(x)$or$Li_2(x)$. How can i do that? 3answers 82 views ### Evaluate the integral.$\int x^2 \log(4x) dx$The problem is$\int x^2 \log(4x) dx$Here$\ln$refers to the natural logarithm. So far, I know$u = x^2$and$du = 2x (dx)$. So$dv = \ln(4x) dx$and$v = 1/x$, but I don't know where to go from ... 1answer 79 views ### Integral of Inverse of Log X What is the value of $$\int\dfrac{1}{\log x}dx$$ I have tried many times, but failed everytime. Can anyone help me out in solving this question. 2answers 394 views ### Integral$\int_0^1\frac{\ln x}{x-1}\ln\left(1+\frac1{\ln^2x}\right)dx$Is it possible to evaluate this integral in a closed form? $$I \equiv \int_{0}^{1}{\ln\left(x\right) \over x - 1}\, \ln\left(1 + {1 \over \ln^{2}\left(x\right)}\right)\,{\rm d}x$$ Numerically, ... 1answer 58 views ### Integration involving$\log_2(x)$Having a hard time going about this problem: $$\int{\frac{\ln(2)\log_2(x)}{x}}$$ I believe$\ln(2)$would be considered a constant, so than the equation would then changed to: ... 2answers 100 views ### How to prove$\sum\limits_{k=2}^{n}\dfrac{1}{k}<\log(n)<\sum\limits_{k=1}^{n-1}\dfrac{1}{k}$How to prove$\sum\limits_{k=2}^{n}\dfrac{1}{k}<\log(n)<\sum\limits_{k=1}^{n-1}\dfrac{1}{k}$It is clear if i consider the area under$f(x)=\dfrac{1}{x})$from$1$to$n$end divide the ... 5answers 370 views ### An integral with irrational exponents$\int_0^\infty\frac{\log\left(\frac{1+x^{4+\sqrt{15}}}{1+x^{2+\sqrt{3}}}\right)}{\left(1+x^2\right)\log x}dx$I was challenged to prove this identity $$\int_0^\infty\frac{\log\left(\frac{1+x^{4+\sqrt{15\vphantom{\large A}}}}{1+x^{2+\sqrt{3\vphantom{\large A}}}}\right)}{\left(1+x^2\right)\log ... 2answers 31 views ### How is natural log integration broken up into this range? (equation is contained the script) When I was reading a paper, I found an strange derivation like$$\int^{+\infty}_{-\infty}\mathrm{ln}(1+e^w)f(w)dw\\=\int^0_{-\infty}\ln(1+e^w)f(w)+\int^\infty_0[\ln(1+e^{-w})+w]f(w)dw$$when w is ... 1answer 374 views ### Find the volume of the solid obtained by rotating the region bounded by y = ln x, y = 0, x = 2 about the x-axis I have the problem: Assuming y = ln(x), and y = 0, find the volume bound by these two lines and the point x = 2 if the area were rotated around the x-axis. I ended up with 2\pi\int_1^2 ... 3answers 87 views ### Difficult Integral Involving the \ln function Please help me solve this integral! I have tried multiple different procedures for integration by parts, as well as substitution and have not come up with anything.$$\int\frac{\ln x}{(\ln x+1)^2}dx$$... 2answers 54 views ### how to find the integral of a rational logarithmic function I can't seem to figure this one out, it is:$$\int\frac{\ln(x)}xdx $$I substituted u for \ln(x), so u = \ln(x) and du = \frac1x dx then to find x in terms of u: e^u = x so ... 2answers 3k views ### U-substitution for integral of 1/(1+e^x)dx. What am I doing wrong? Here is my work, witth the right answer. I feel like every step is right, but somehow I am getting the wrong answer. How?$$ \int \frac{1}{1+e^z}dz = \int\frac{1}{e^z(\frac{1}{e^z} + 1)}dz ... 1answer 49 views ### Integral of$\log(1+x^{-2})$How can I find the integral of this function:$f(x) =\int{\log(1+\frac{1}{x^2})}dx$What technique should I use? 2answers 81 views ###$\sum_{i=1}^n 1/i \leq c\log n$This is what I want to show:$\sum_{i=0}^n 1/i \leq c \log n$for all$n>N$My current approach was this:$\sum_{i=1}^n 1/i = ( \int \sum_{i=1}^n 1/i )' = ( \sum_{i=1}^n \int 1/i )' = ( ...
A friend asked me what was the solution to the problem (which was on his test)$$\int\frac4xdx$$ I proceeded to tell him that you can take out the 4 in the numerator, and then just take the integral of ...