2
votes
3answers
80 views

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 ...
8
votes
2answers
151 views

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} ...
9
votes
2answers
311 views

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 ...
1
vote
1answer
34 views

Logarithmic integral and natural numbers.

Prove these two relations: $$\text{li}(k+1)+k-\log (k)-\gamma = \int_0^k \left(\int_1^2 \frac{(s+1)^{n-1}+s-1}{s} \, dn\right) \, ds$$ $$n = \lim_{s\to 0} \, \frac{(s+1)^{n-1}+s-1}{s}$$ ...
7
votes
1answer
194 views

An equivalent for $\int_0^1\left(\frac{1}{\log x}+\frac{1}{1-x}\right)^n\;dx$

Set $$ I_n :=\int_0^1\left(\frac{1}{\log x} + \frac{1}{1-x}\right)^n \:\mathrm{d}x \qquad n=1,2,3,.... $$ We have $$I_1 =\gamma, \quad I_2 =\log (2 \pi) - \frac 32, \quad I_3 = 6 \log A - ...
1
vote
2answers
41 views

Proof that $\log(a^b) = b\log a$ when $\log$ is defined by an integral

When $\log a$ is defined as $\displaystyle\int_1^a\frac{dx}x$, then how does one prove that $\log(a^b)=b\log a$? I will post an answer here that is identical to an answer I posted to another ...
8
votes
2answers
216 views

A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$

I met this integral $$ \int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{ \mathrm{d}x}{1-x} \qquad (*) $$ while evaluating this log-cosine integral. I made several ...
3
votes
3answers
103 views

$\int_{0}^{\pi/2}\ln\left(1+4\sin^4 x\right)\mathrm{d}x$ and the golden ratio

We already know that, for any real number $t$ such that $t\geq-1$, $$ \int_{0}^{\pi/2} \ln \left(1+t \sin^2 x\right) \mathrm{d}x = \pi \ln \left( \frac{1+\sqrt{1+t}}{2} \right). $$ Prove that ...
4
votes
1answer
90 views

Prove this polylogarithmic integral has the stated closed form value

Question. Prove the following polylogarithmic integral has the stated value: $$I:=\int_{0}^{1}\frac{\operatorname{Li}_2{(1-x)}\log^2{(1-x)}}{x}\mathrm{d}x=-11\zeta{(5)}+6\zeta{(3)}\zeta{(2)}.$$ ...
2
votes
4answers
135 views

What is the integral of x/ln(x)?

Well, I'm french so excuse me if I make some mistakes in english... I have to calculate this integral : $$ \int_{e}^{2e} \frac{x}{\ln(x)} dx $$ But I don't know how, can you help me please? Thank ...
14
votes
4answers
560 views

A closed form for $\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x$

We already know that \begin{align} \displaystyle & \int_{0}^{\pi/2} x \ln(2 \cos x)\:\mathrm{d}x = -\frac{7}{16} \zeta(3), \\\\ & \int_{0}^{\pi/2} x^2 \ln^2(2 \cos x)\:\mathrm{d}x = ...
4
votes
0answers
131 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting two rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos ...
5
votes
3answers
178 views

Help with logarithmic definite integral: $\int_0^1\frac{1}{x}\ln{(x)}\ln^3{(1-x)}$

I'm look for a closed form evaluation of the following improper definite integral involving logarithms: $$\begin{align} I:&=\int_{0}^{1}\frac{1}{x}\ln{(x)}\ln^3{(1-x)}\,\mathrm{d}x\\ ...
3
votes
3answers
80 views

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 ...
2
votes
1answer
128 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 ...
0
votes
2answers
87 views

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.
2
votes
0answers
44 views

Log Log Integrals II

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) - ...
7
votes
4answers
361 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: $$ \int_0^1\frac{x^{19}-1}{\ln x}\,dx $$
0
votes
2answers
47 views

Integral of $e^x ln(e^{2x} - 4)$

Find the integral from ln4 to ln6 of $$e^x \ln(e^{2x} - 4)$$ 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} + ...
0
votes
2answers
63 views

A definite integral involve Logarithmic Functions

Here is the integral body: $$\int_0^m {{x^a}\ln \left( {x + b} \right)dx,a > - \frac{3}{2},b > 0} $$
1
vote
1answer
21 views

Help with integral/logarithm inequality

I have to prove the following inequality: $1/(n+1) < \int_n^{n+1} 1/t$ $dt$ $<1/n$ I thought it would be easier to attack this via integration, so I get: $1/(n+1) <$ log $(n+1)-$ ...
18
votes
2answers
370 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, ...
28
votes
5answers
345 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 ...
2
votes
1answer
37 views

Definite integral of partial fractions?

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} + ...
5
votes
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 ...
2
votes
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 ...
0
votes
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 ...
33
votes
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 ...
26
votes
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 ...
25
votes
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 ...
20
votes
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 ...
24
votes
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 ...
27
votes
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? $$\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}dx=\frac{\pi^2}8-\frac12$$
17
votes
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: $$\large\int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx$$
20
votes
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 ...
21
votes
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? $$\int_0^\infty\frac{1}{\sqrt[3]{x}}\left(1+\log\frac{1+e^{x-1}}{1+e^x}\right)dx$$
9
votes
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 ...
2
votes
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 ...
22
votes
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 ...
3
votes
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.
0
votes
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 ...
19
votes
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) ...
28
votes
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 ...
34
votes
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 ...
1
vote
1answer
827 views

Another two hard integrals

Evaluate : $$\begin{align} & \int_{0}^{\frac{\pi }{2}}{\frac{{{\ln }^{2}}\left( 2\cos x \right)}{{{\ln }^{2}}\left( 2\cos x \right)+{{x}^{2}}}}\text{d}x \\ & \int_{0}^{1}{\frac{\arctan ...
2
votes
1answer
423 views

Definite integral involving e and ln

I am supposed to solve a definite integral which involves $\ln$ and $e$. Nowhere in my textbook can I even find examples of how this would be done. I do know that $\ln(e(x))$ and $e(\ln(x))$ ...
1
vote
1answer
163 views

How to calculate $\int_{|z|=r}\ln(1-z)\,dz$ in dependence of $r\neq1$?

With the integration I mean one counter-clockwise turn around the origin, i.e. $$\int_{\phi=0}^{2\pi}\ln(1-re^{i\phi})ire^{i\phi}d\phi$$ For $r<1$, this is simply a contour integration on a ...
3
votes
4answers
566 views

How to evaluate $\int_{1}^{2}\frac{dx}{1+x+\ln x}$?

Can you help me find the value of the integral $$\int_{1}^{2}\frac{dx}{1+x+\ln x}$$ Thank you
3
votes
3answers
145 views

Squeeze an integral

Would you have any idea about this problem ? Prove that for all nonnegative integers $n$, the following inequalities hold: $$\frac{e^2}{n+3}\leq \int_{1}^{e} x (\ln x)^n \,dx \leq ...