All Questions
Tagged with closed-form calculus
835
questions
2
votes
1
answer
99
views
Integrating $\int_{0}^{\infty} \frac{\tanh \left(\frac{x}{2}\right)+\tanh (2 x)}{x}\left(e^{\frac{3 x}{2}}-1\right)^{2} e^{-4 x} \mathrm{~d}x$
Show that
$$\int_{0}^{\infty} \frac{\tanh \left(\frac{x}{2}\right)+\tanh (2 x)}{x}\left(e^{\frac{3 x}{2}}-1\right)^{2} e^{-4 x} \mathrm{~d}x=\boxed{2\log\left(\frac{768\sqrt2\pi^4}{25\Gamma^3\left(\...
4
votes
2
answers
211
views
How to calculate this integral $\int_{0}^{\infty} \left(\frac{x - 1}{\ln(x)}\right)^2 \cdot \frac{1}{1 + x^n} \,dx$
How to calculate this integral
$$ \int_{0}^{\infty} \left(\frac{x - 1}{\ln(x)}\right)^2 \cdot \frac{1}{1 + x^n} \,dx$$
This is how I start
$$f(a)=\int_{0}^{\infty} \frac{(x^a-1)^2}{\ln^2(x)}\frac{1}{...
5
votes
3
answers
461
views
Integration of $ \int_{0}^{\frac{\pi}{2}} x \log(1-\cos x) \,dx $ [closed]
Question: What is the closed form of this following integral? $
\int_{0}^{\frac{\pi}{2}} x \log(1-\cos x) \,dx.$
Here is my solution
we know that
$$\displaystyle{\sum\limits_{n = 1}^\infty {\frac{{\...
2
votes
2
answers
227
views
Evaluate the infinite product $ \prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right )$
Question statement
Evaluate the infinite product
$$\displaystyle{\prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right ) }$$
My try
Because of the square of $\displaystyle{x}$ , we can consider $...
5
votes
5
answers
376
views
How to evaluate $\int_{-\infty}^{+\infty}\frac{\cos x}{\left(1+x+x^2\right)^2+1}\mathrm{~d}x$
Question
$$\int_{-\infty}^{+\infty}\frac{\cos x}{\left(1+x+x^2\right)^2+1}\mathrm{~d}x$$
Wolfram alpha says it is
$$\int_{-\infty}^{\infty} \frac{\cos(x)}{\left(1 + x + x^2\right)^2 + 1} \,dx = \frac{\...
2
votes
0
answers
141
views
closed form for limit?
Consider the function
$$ f(x)=\lim_{k \to \infty}\bigg(\int_0^x \sum_{n=1}^k e^{\frac{\log n}{\log r}}~dr \bigg)\bigg( \int_0^1 \sum_{n=1}^k e^{\frac{\log n}{\log r}}~dr \bigg)^{-1} $$
I want to find ...
- 756
2
votes
1
answer
151
views
For $J=\{1,2,\dots,n \}$ is there an easy way to compute $\prod\limits_{i\in J | i \ne k} (k-i)$?
When I studied calculus at my university there is one question that I hated the most which is given a finite number of terms for some sequence find the $n-$th term. I hated this type of question ...
- 6,620
3
votes
4
answers
276
views
$\int_{-1}^u\text{exp}(\frac{1}{x^2-1})dx$
I'm trying to compute $\int_{-1}^u\text{exp}(\frac{1}{x^2-1})dx$ where $u\in[-1,1]$.
This is a crucial element of this paper and I need to be able to compute it quickly in Mathematica thousands of ...
- 303
2
votes
0
answers
67
views
Closed form for $\psi^{1/k}(1)$, where $k$ is an integer
I have proven the identity
$$
\sum_{k=1}^{\infty} \dfrac{\operatorname{_2F_1}(1, 2, 2-1/t,-1/k)}{{k}^{2}} = Γ(2-\dfrac{1}t){\psi^{1/t}(1)}+\psi(-\dfrac{1}t)(\dfrac{1}t(1-\dfrac{1}t))+\gamma(1-\dfrac{1}...
6
votes
6
answers
378
views
Show that $\int_0^1\int_{1-y}^1\sqrt{(x-1)(y-1)(x+y-1)}\mathrm dx\mathrm dy=\frac{2\pi}{105}$.
How can we show that $\int_0^1\int_{1-y}^1\sqrt{(x-1)(y-1)(x+y-1)}\mathrm dx\mathrm dy=\frac{2\pi}{105}$ ?
Desmos says it's true.
The inner indefinite integral is not nice. And strangely, when I plug ...
- 25.7k
1
vote
0
answers
42
views
Analytical Solution for a Double Integral Involving Logistic Functions and Gaussian Distributions
I am working on a mathematical problem involving the evaluation of a double integral, and I am seeking an analytical solution or techniques to solve it. The integral I'm dealing with is as follows:
$$...
- 23
1
vote
0
answers
59
views
Primitive of $e^\frac{-1}{1-x^2}$
I'm currently reading a paper where they construct a $C^\infty$ function $\varphi$ on $\mathbb{R}$ that is symmetric and decreasing on $\mathbb{R}^+$ supported on $[-1,1]$ such that $0\leq \varphi(x)\...
- 303
3
votes
1
answer
131
views
Show that $\int_{0}^{1} \frac{\log(1 - x^2)}{\sqrt{x} (\sqrt{x} + 1)} \,dx = \frac{7}{2} \ln(2) - \frac{5}{4} \zeta(2)$
$$\int_{0}^{1} \frac{\log(1 - x^2)}{\sqrt{x} (\sqrt{x} + 1)} \,dx = \frac{7}{2} \ln(2) - \frac{5}{4} \zeta(2)$$
Here is my try
\begin{align*}
\text{Let: } & t = \sqrt{x} \Rightarrow dt = \frac{dx}{...
5
votes
1
answer
125
views
A challenging Integral Involving Logarithmic and Trigonometric Functions
Question: How to evaluate $$\frac{1}{2\sqrt{2}} \int_{0}^{\frac{\pi}{2}} \frac{\log(1 + \tan y)}{(\cos y + \sqrt{2} \sin(y + \frac{\pi}{4})) \sqrt{1 + \sqrt{2} \sin(2y + \frac{\pi}{4})}} \, dy = G$$
...
2
votes
0
answers
70
views
Closed form for ${_3F_2}\!\left(\begin{array}c\tfrac34,1,1\\\tfrac32,\tfrac74\end{array}\middle|1\right)$
I am trying to find the closed form the expression $${_3F_2}\!\left(\begin{array}c\tfrac34,1,1\\\tfrac32,\tfrac74\end{array}\middle|1\right).$$ I was able to convert the expression into the series $${...
