0
votes
0answers
45 views

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 ...
2
votes
2answers
124 views

$\int_0^1\frac{1-t}{(t-2)\ln t}\,dt$ 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 ...
8
votes
0answers
152 views
+50

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
2
votes
3answers
95 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 ...
0
votes
0answers
29 views

Integrating the logarithm of a function including a square root of a second degree polynomial

I have been trying for some time to calculate the following integral: $$\int \ln\left(k+\sqrt{ax^2+bx+c}\right)\ dx$$ where k, a, b and c are real numbers. I have tried several strategies, but without ...
8
votes
2answers
167 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} ...
11
votes
2answers
353 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 ...
7
votes
1answer
216 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 - ...
2
votes
2answers
56 views

Integrating 1/x

The standard definition of integrating $\frac{1}{x}$ is: $$ \int \frac{dx}{ax + b} = \frac {1}{a} \ln |ax + b| + K $$ Now, if I'm understanding the "constant factor rule", that is: $$ \int k ...
8
votes
2answers
232 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
123 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
109 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
136 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 ...
15
votes
4answers
585 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
140 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 ...
2
votes
1answer
99 views

Calculate $I= \int_{1}^{e}\frac{(1+\ln x)x}{(1+x\ln x)^2}dx$

Please help me solve this: (level = high school) $$ \int_{1}^{e}\frac{(1+\ln x)x}{(1+x\ln x)^2}\,dx $$ Thanks
7
votes
3answers
205 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
84 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 ...
1
vote
1answer
46 views

$\sum_{x=a}^{b-1}\frac{1}{x}$ and $\sum_{x=a+1}^b\frac{1}{x}$

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} ...
2
votes
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 ...
1
vote
2answers
71 views

Commutation between Logarithm and Gaussian Integral.

I'm calculating a partition function (physics) and I arrive to the following expression: $$\log \int_{-\infty}^{\infty} \frac{du}{\sqrt{2\pi}} e^{-u^2/2} e^{-nq/2}[2\cosh(\sqrt{q}\,u+m)]^n \qquad(1)$$ ...
1
vote
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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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 ...
7
votes
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: $$ \int_0^1\frac{x^{19}-1}{\ln x}\,dx $$
5
votes
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. ...
0
votes
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 ...
4
votes
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} ...
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} + ...
2
votes
0answers
53 views

Integral of Difference of Logs

I get the expansion of $h$ to be $$ h(z) = {1 \over z } \sum_{r=1}^{\infty}{1 \over r}{(-{\alpha \over z}})^r $$ $$ \Rightarrow h(z) = \sum_{r=-2}^{-\infty}{{(-\alpha)^{r+1} \over -(r+1)} z^{r}} $$ ...
3
votes
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?
1
vote
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 ...
0
votes
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.
18
votes
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, ...
0
votes
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: ...
0
votes
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 ...
30
votes
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 ...
1
vote
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 ...
2
votes
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 ...
4
votes
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. $$\int\frac{\ln x}{(\ln x+1)^2}dx$$ ...
1
vote
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 ...
5
votes
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 ...
2
votes
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?
0
votes
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 )' = ( ...
1
vote
2answers
65 views

General solution to the integral of 4/x

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 ...
0
votes
1answer
86 views

Undoing the Natural log after integrating $ln \frac{ \sec{x} \tan{x}}{3x+5}dx $

Since beating my head against a brick wall is so fun, I kept working on this old integral $\int \frac{ \sec{x} \tan{x}}{3x+5}dx$ . I think I have finally found a way to do it. Here goes. $$ \int ...
0
votes
2answers
64 views

Are the sums equal to each other?

They are $2$ different results for the integral $$\int xe^{2x}\sin\left(\frac x3\right)\,dx$$ $\displaystyle\frac{-3}{1369}e^{2x}\left(3(35-74x)\sin\left(\frac x3\right)+(37x-36)\cos\left(\frac ...
1
vote
1answer
58 views

Growth of |logx| versus of 1/x

Do you think there is a number k s.t. $\int_{(0,\infty)} \frac{|log(x)|^{k}}{x}d\mu$ will converge,where $\mu$ is the Lebesgue measure? If you don't know ,can you at least give me some reference for ...
5
votes
2answers
248 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 ...