All Questions
Tagged with closed-form calculus
835
questions
9
votes
1
answer
384
views
evaluation of $\sum_{n=1}^{\infty} (-1)^{n-1} \frac{H_{n} H_{n+1}^{(2)}}{(n+1)^{2}}$ and other Euler sums
I was trying to evaluate this famous integral $$\int_{0}^{1} \frac{\ln (x) \ln^{2}(1+x) \ln(1-x)}{x} \ dx $$
Here is my attempt so solve the integral
\begin{align}
&\int_{0}^{1} \frac{\ln (x) \ln^{...
user1313975
7
votes
1
answer
295
views
how to evaluate $\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x$
Question: how to evaluate $$\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x$$
MY try to evaluate the integral
$$
\begin{aligned}
& I=\int_0^{\infty} \frac{x \...
user1313975
7
votes
3
answers
228
views
Closed form for $\int_0^{\pi/2}\arctan\left(\frac12\sin x\right)\mathrm dx$?
Is there a closed form for $I=\int_0^{\pi/2}\arctan\left(\frac12\sin x\right)\mathrm dx$ ?
Context
Earlier I asked "Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\...
- 25.7k
2
votes
1
answer
78
views
How to evaluate $\int_1^{\infty}\frac{t^2\ln^2 t\ln(t^2-1)}{1+t^6}{\rm d}t $
I was evaluating Evaluate $\displaystyle\int_0^{\infty} x^2\ln(\sinh x)\operatorname{sech}(3 x){\rm
d}x .$
On the path of integrating the main function, I am stuck at this integral. I don't know how ...
4
votes
3
answers
220
views
Show that $\int_{0}^{1} \frac{\tan^{-1}(x^2)}{\sqrt{1 - x^2}} \, dx = \frac{1}{2}\pi \tan^{-1}\left(\sqrt{\frac{1}{\sqrt{2}} - \frac{1}{2}}\right)$
Problem: Show that $$\int_{0}^{1} \frac{\tan^{-1}(x^2)}{\sqrt{1 - x^2}} \, dx = \frac{1}{2}\pi \tan^{-1}\left(\sqrt{\frac{1}{\sqrt{2}} - \frac{1}{2}}\right)$$
Some thinking before trying
At least we ...
12
votes
1
answer
653
views
Prove $\int_0^\pi\arcsin(\frac14\sqrt{8-2\sqrt{10-2\sqrt{17-8\cos x}}})dx=\frac{\pi^2}{5}$.
There is numerical evidence that
$$\int_0^\pi\arcsin\left(\frac14\sqrt{8-2\sqrt{10-2\sqrt{17-8\cos x}}}\right)dx=\frac{\pi^2}{5}.$$
How can this be proved?
Context
In another question, three random ...
- 25.7k
4
votes
2
answers
238
views
Closed form for this generalisation of the gamma function. $f(x+1)=f(x)g(x+1) $
Just for curiosity I want to generalise the Pi function i.e $f(x+1) = f(x)g(x+1)$ for some differentiable function, I know this function probably has no closed form for general functions $g$ as I ...
- 6,620
20
votes
1
answer
1k
views
Prove $\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12$.
There is numerical evidence that
$$I=\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12.$$
How can this be proved?
...
- 25.7k
2
votes
0
answers
247
views
Is it possible to evaluate $\int_{0}^{\frac{\pi}{2}} e^{-(\pi \tan(x) - 1)^2} \, dx$?
How to evaluate $$\int_{0}^{\frac{\pi}{2}} e^{-(\pi \tan(x) - 1)^2} \, dx$$
Source: I created this integral so I don’t know the closed form
I tried Wolfram Alpha, but Wolfram Alpha is unable to ...
6
votes
0
answers
104
views
Please help me identify any errors in my solution to the following DE: $xf(x)-f'(x)=0$, $f(0)=1$
Context/background:
I am self-studying series, first in the context of generating functions and now in the context of functional/differential equations. As such, I like to set myself practise problems,...
- 1,632
14
votes
1
answer
637
views
Show that $\int_{\arccos(1/4)}^{\pi/2}\arccos(\cos x (2\sin^2x+\sqrt{1+4\sin^4x})) \mathrm dx=\frac{\pi^2}{40}$
There is numerical evidence that $$I=\int_{\arccos(1/4)}^{\pi/2}\arccos\left(\cos x\left(2\sin^2x+\sqrt{1+4\sin^4x}\right)\right)\mathrm dx=\frac{\pi^2}{40}$$
How can this be proved?
I was trying to ...
- 25.7k
0
votes
1
answer
120
views
How to integrate $\int_{2}^{\infty} \frac{\pi(x) \ln(x^{\sqrt{x}}) \cdot (x^2 + 1)}{(x^2 - 1)^3} \,dx$
How to integrate $$\int_{2}^{\infty} \frac{\pi(x) \ln^2(x^{\sqrt{x}}) \cdot (x^2 + 1)}{(x^2 - 1)^3} \,dx \quad?$$
Wolfram gives the numerical value
$$\int_{2}^{\infty} \frac{\pi x (1 + x^2) \log^2(x^{\...
3
votes
1
answer
241
views
Remarkable logarithmic integral $\int_0^1 \frac{x \log ^2(x) \log (1-x)}{1+x^2} dx$
Question: how to evaluate this logarithm integral?
$$
I=\int_0^1 \frac{x \log ^2(x) \log (1-x)}{1+x^2} d x
$$
My attempt:
$$
\begin{aligned}
I=&\int_0^1 \frac{x \log ^2(x) \log (1-x)}{1+x^2} d x\\
...
3
votes
1
answer
183
views
Improper Integral $\int_{0}^{\infty} \log(t) t^{-\frac{1}{2}} \exp\left\{-t\right\} dt$
Background
Hi. I am currently writing my undergraduate thesis which mainly revolves around the generalized log-Moyal distribution pioneered by Bhati and Ravi (see here). In the aforementioned article, ...
- 399
2
votes
2
answers
148
views
How to evaluate $\int_{0}^{1} \int_{0}^{1} \frac{{(1 + x) \cdot \log(x) - (1 + y) \cdot \log(y)}}{{x - y}} \cdot (1 + \log(xy)) \,dy \,dx$
Question: How to evaluate this integral $$\int_{0}^{1} \int_{0}^{1} \frac{{(1 + x) \cdot \log(x) - (1 + y) \cdot \log(y)}}{{x - y}} \cdot (1 + \log(xy)) \,dy \,dx$$
My messy try
$$\int_{0}^{1} \int_{...