All Questions
Tagged with logarithms integration
740
questions
4
votes
2
answers
287
views
Showing $\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right) \, dx=\frac{\pi^2 + 4}{2}$
While exploring possible applications for exponential substitution, I stumbled upon the following integral identity:
$$\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right)...
- 1,011
3
votes
3
answers
385
views
$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ as a limit of a sum
Working on the same lines as
This/This and
This
I got the following expression for the Dilogarithm $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$:
$$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
- 862
3
votes
3
answers
103
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Why to use modulus in integration of $1/x$ [closed]
$$ \int \frac1x = \log_e |x|+C$$
Why is modulus sign needed. If this is because the domain of logarithmic function is $(0,\infty)$ Then why don't we mention the limitations of the domains of other ...
- 199
1
vote
1
answer
44
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How to prove upper bound of this difference of the Sine Integral?
This exercise can be found in Mathematics LibreTexts (bottom of the page) . I have been stuck for about a day and have made minimal progress.
Let $S(x)=\int_0^x\frac{\sin t}{t}$.
Show that for $k \ge ...
1
vote
1
answer
54
views
Logarithmic Function Calculation in Mathematica
I find these results in the evaluation of the logarithms that only differ in the sign $-$ I do not understand why in the first case $\operatorname{Log}[x+1]/8$ is not returned as an answer.
2
votes
0
answers
100
views
Is ln|x| + C really the most general antiderivative of 1/x? [duplicate]
I recently stumbled across a claim that $\ln |x| + C$ isn't the most general antiderivative of $1/x$. The argument was that the parts of the curve $\ln |x|$ separated by the $y$-axis do not have to be ...
- 508
1
vote
2
answers
81
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Is there a closed form for the integral $\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\mathrm{d}z$?
As the title says, I would like to know if there is a closed form for the integral:
\begin{align*}
\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\...
- 31
0
votes
1
answer
58
views
The same equation giving different integrals?
I feel like I’m missing something obvious. I have checked on online integral calculators and I keep getting different answers despite the fact they are equivalent fractions.
$$\frac{1}{0.5x+5}=\frac{2}...
- 1
8
votes
1
answer
177
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how to integrate $\int_0^1 \ln^4(1+x) \ln(1-x) \, dx$?
I'm trying to evaluate the integral $$\int_0^1 \ln^4(1+x) \ln(1-x) \, dx,$$ and I'd like some help with my approach and figuring out the remaining steps.
or is it possible to evaluate $$\int_0^1 \ln^n(...
3
votes
1
answer
146
views
how to evaluate this integral $\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$
Question statement: how to evaluate this integral $$\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$$
I don't know if there is a closed form for this integral or not.
Here is my attempt to solve ...
2
votes
2
answers
184
views
Closed form of the integral $\int_{0}^{1} \log^n \left (\frac {1-x}{1+x}\right )dx$
I found this nice integral
$$i=\int_{0}^{1} \log^3\left (\frac {1-x}{1+x}\right)\;dx\tag{1}$$
on youtube but I don't remember where.
Let us generalize a bit to a power $n=0, 1, 2, ...$ and ask for the ...
- 12.7k
2
votes
1
answer
79
views
Shouldn't the integral of $1/x$ be $\text{sgn}(x)\ln|x|$?
The integral of $1/x$ across an interval $[-b,-a]$ would be the negative of the integral across the interval $[a,b]$, no? So, the equation ought to be:
$$\int \frac 1x = \text{sgn}(x)\ln|x| + C$$
...
- 1,129
1
vote
2
answers
123
views
How to integrate $\int_{0}^{1} \frac{x \operatorname{Li}_2(1 - x)}{1 + x^2} \, dx$
How to integrate $$\int_{0}^{1} \frac{x \operatorname{Li}_2(1 - x)}{1 + x^2} \, dx$$
My try to integrate
$$\text{I}=\int_{0}^{1} \frac{x \operatorname{Li}_2(1 - x)}{1 + x^2} \, dx$$
\begin{aligned}
&...
2
votes
2
answers
118
views
does Integrating both sides of an equation in dx will Invalidates the equality?
I'm struggling to grasp the justification behind integrating both sides of an equation. While I understand that operations can be applied to both sides, maintaining equality, it appears that this ...
- 43
0
votes
0
answers
50
views
How to integrate $\frac{x^N\log(1+x)}{\sqrt{x^2+x_1^2}\sqrt{x^2+x_2^2}}$?
I am trying to compute the integral
$$\int_{x_0}^{1}\frac{x^N\log(1+x)}{\sqrt{x^2+x_1^2}\sqrt{x^2+x_2^2}}\text{d}x$$
where $x_0, x_1$ and $x_2$ are related to some parameters $\kappa_\pm$ by
$$x_0=\...
- 1