All Questions
81
questions
2
votes
1
answer
177
views
Show that $ \int_0^{\pi\over 2}\frac{\sin(2nx)}{\sin^{2n+2}(x)}\frac{1}{e^{2\pi \cot x}-1}dx =(-1)^{n-1}\frac{2n-1}{4(2n+1)} $
Show that $$ \int_0^{\pi\over 2}\frac{\sin(2nx)}{\sin^{2n+2}(x)}\frac{1}{e^{2\pi \cot x}-1}dx =(-1)^{n-1}\frac{2n-1}{4(2n+1)}
$$
My attempt
Lemma-1
\begin{align*}
\frac{\sin(2nx)}{\sin^{2n}(x)}&=\...
0
votes
0
answers
132
views
Calculation of $\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$
Calculation of $$\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$$
My attempt
\begin{align*}
\sum_{n=1}^\infty\frac{\psi_1(n)}{2^n n^2} &= -\sum_{n=1}^\infty\psi_1(n)\left(\frac{\log(2)}{2^n n}+\int_0^...
8
votes
2
answers
244
views
How to calculate $\int _0^1 \int _0^1\left(\frac{1}{1-xy} \ln (1-x)\ln (1-y)\right) \,dxdy$
Let us calculate the sum
$$
\displaystyle{\sum_{n=1}^{+\infty}\left(\frac{H_{n}}{n}\right)^2},
$$
where $\displaystyle{H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}}$ the $n$-th harmonic number.
My try
The ...
1
vote
1
answer
95
views
Evaluation of $\int_{0}^{\frac{\pi}{4}} \frac{\log(\log(\tan(\frac{\pi}{4} + x))) \cdot \log(\tan(\frac{\pi}{4} + x)))}{\tan(2x)} \,dx$ [closed]
$$\int_{0}^{\frac{\pi}{4}} \frac{\log(\log(\tan(\frac{\pi}{4} + x))) \cdot \log(\tan(\frac{\pi}{4} + x)))}{\tan(2x)} \,dx$$
$$\int_{0}^{\frac{\pi}{4}} \frac{\log(\log(\tan(\frac{\pi}{4} + x))) \cdot \...
5
votes
2
answers
157
views
Show that $\sum_{n=1}^{\infty} \frac{\binom{2n}{n} (H_{2n} - H_n)}{4^n (2n - 1)^2} = 2 + \frac{3\pi}{2} \log(2) - 2G - \pi$
Show that $$\sum_{n=1}^{\infty} \frac{\binom{2n}{n} (H_{2n} - H_n)}{4^n (2n - 1)^2} = 2 + \frac{3\pi}{2} \log(2) - 2G - \pi$$
My try :
We know that
$$\sum_{n=1}^{\infty} \binom{2n}{n} (H_{2n} - H_{n}) ...
3
votes
2
answers
206
views
why does $\pi$ always show up in $\int_0 ^1 \frac{x^c}{1+x^k} dx$ if $c\neq mk-1$ for all $m \in \mathbb{N}$
when I posted this question I was interested in the sum $ \sum_{n=0} ^{\infty} \frac{(-1)^n}{4n+3}$ but when I thought about the generalised sum $ \sum_{n=0} ^{\infty} \frac{(-1)^n}{kn+c +1}$ for all $...
- 6,620
11
votes
3
answers
451
views
How to evaluate $ \sum\limits_{k=0} ^{\infty} \frac{(-1)^k}{4k+3}$?
I was trying to solve the integral $\int_0 ^{\frac{\pi}{4}} \sqrt{\tan{x}}dx$ and I noticed I can do the following:
$$\int_0 ^{\frac{\pi}{4}} \sqrt{\tan{x}}dx=\int_0 ^{\frac{\pi}{4}} \sqrt{\tan{x}} \...
- 6,620
5
votes
2
answers
185
views
Generating Function $\sum_{k=1}^{\infty}\binom{2k}{k}^{-2}x^{k}$
Closed Form For :
$$S=\sum_{k=1}^{\infty}\binom{2k}{k}^{-2}x^{k}$$
Using the Series Expansion for $\arcsin^2(x)$ one can arrive at :
$$\sum_{k=0}^{\infty}\binom{2k}{k}^{-1}x^{k}=\frac{4}{4-x}-4\arcsin\...
- 3,280
1
vote
0
answers
60
views
Integration including the floor function
I'm aware of the way to quantify the following integral in the following way, however I'm trying to find another way to express the given integral. Especially when the function $f(x)$ is not ...
- 73
2
votes
3
answers
124
views
Evaluating $\int_{0}^{1}{\tan^{-1}(x)\, dx}$ via a series (not IBP)
I tried :
$$\begin{align}I&=\int_{0}^{1}{\tan^{-1}(x)\, dx}\\&=\int_{0}^{1}{\sum_{k\geq0}{\frac{(-1)^kx^{2k+1}}{2k+1}}\, dx}\\&=\sum_{k\geq0}{\frac{(-1)^k}{2k+1}\left[\frac{x^{2(k+1)}}{2(k+...
- 977
4
votes
0
answers
135
views
Simplify a summation in the solution of $\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x$
Context
I calculated this integral:
$$\begin{array}{l}
\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x=\\
\displaystyle\frac{n!}{c^{n+1}}\left\lbrace\sum_{k=0}^{n}\left[\text{Ci}\left(...
3
votes
1
answer
129
views
Compact form of solution of $\displaystyle\int_0^{\frac{\pi}{2}}\ln\left(1+\alpha^n\sin(x)^{2n}\right)\mathrm{d}x$
I hope it won't be categorized as a trivial question, I solved this integral and arrived at the following form:
$${\int_{0}^{\frac{\pi}{2}}\ln\left(1+\alpha^n\sin(x)^{2n}\right)\mathrm{d}x=\frac{n\pi}{...
1
vote
1
answer
153
views
Sum of values across a line?
I've been thinking recently and have confused myself a little bit as to the difference between sums and integrals.
I understand that an integral is a Riemann sum where you take the limit as the ...
- 13
7
votes
3
answers
292
views
Solve $\int_0^\infty\frac x{e^x-e^{\frac x2}}dx$
I was able to solve the integral $$\int_0^\infty\frac x{e^x-e^\frac x2}dx=4\left(\frac{\pi^2}6-1\right)$$ I want to see other approaches to solving it. Here is my solution: $$\int_0^\infty\frac x{e^x-...
- 6,549
7
votes
3
answers
277
views
Evaluating a Logarithmic Integral
For everything on this post $n$ and $m$ are positive integers.
The other day I found the following integral on the popular post "Integral Milking" and decided to give it a go.
$$\large\int_{...
2
votes
0
answers
84
views
Is it possible to compute this integral using this double summation?
I attempted to evaluate the following integral, $$\int_{0}^{1} \frac{\ln (1+x)}{1+x^2}\, dx$$
although I know it can be evaluated using Feynman's Technique of Integration, Trigonometric Substitution, ...
- 423
2
votes
2
answers
212
views
$\int_0^1\frac{1}{7^{[1/x]}}dx$
$$\int_0^1\frac{1}{7^{[1/x]}}dx$$
Where $[x]$ is the floor function
now as the exponent is always natural, i converted it to an infinite sum
$$\sum\limits_{k=1}^{\infty} \frac{1}{7^{[1/k]}}$$
Which is ...
- 1,346
1
vote
1
answer
254
views
Prove that $\sum_{k=1}^n 1/k - \ln n \to \int_0^1 1/(1-x) + 1/(\ln x) dx$ as $n\to\infty$.
Prove that
$$
\sum_{k = 1}^{n}\frac{1}{k} - \ln\left(n\right) \to
\int_{0}^{1}\left[\frac{1}{1 - x} +
\frac{1}{\ln\left(x\right)}\right]{\rm d}x
\quad\mbox{as}\quad n\to\infty
$$
I don't think this ...
- 2,031
0
votes
0
answers
46
views
Why can we approximate a sum by a definite integral?
From wikipedia https://en.wikipedia.org/wiki/Summation#Approximation_by_definite_integrals, I read that
$\int_{s=a-1}^{b} f(s)\ ds \le \sum_{i=a}^{b} f(i) \le \int_{s=a}^{b+1} f(s)\ ds$ for increasing ...
- 365
2
votes
1
answer
175
views
Infinite summation of recursive integral
Let $I_n=\int_{0}^{1}e^{-y}y^n\ dy$, where $n$ is non-negative integer. Find $\sum_{n=1}^{\infty}\frac{I_n}{n!}.$
I first solved $I_n$ and obtained $$I_n=-\frac{1}{e}+nI_{n-1} \\
\hspace{35mm} =-\...
- 175
0
votes
0
answers
95
views
Proof of $sin$ formula.
I am reading this quesiton and accepted answer.
Question is about proof.
$S = \sin{(a)} + \sin{(a+d)} + \cdots + \sin{(a+nd)}$
$S \times \sin\Bigl(\frac{d}{2}\Bigr) = \sin{(a)}\sin\Bigl(\frac{d}{2}\...
- 879
2
votes
2
answers
334
views
Is the Beukers-Kolk-Calabi substitution incorrect?
Consider the sum following sum:
$$ I=\sum_{i = 0}^{\infty}\frac{(-1)^{i}}{(2i+1)^{2}(2i+2)}. $$
Clearly this can be transformed into a triple integral:
$$ I=\int_{0}^{1}\int_{0}^{1}\int_{0}^{1} \frac{...
- 1,037
2
votes
1
answer
107
views
Find the value of $\int_0^1{4dx\over 4x^2-8x+3}$
Find the value of $\displaystyle\int_0^1{4dx\over 4x^2-8x+3}$
$$\begin{align*}\int_0^1{4dx\over 4x^2-8x+3}&=\int_0^1{dx\over (x-1)^2-(\frac 12)^2}
\\&=\int_0^1{dx\over (x)^2-(\frac 12)^2}
\\&...
- 3,396
2
votes
3
answers
146
views
Computing $\int_{-2}^{2}\frac{1+x^2}{1+2^x} dx$
I am trying to compute the following integral by different methods, but I have not been able to come up with the result analytically.
$$\int_{-2}^{2}\frac{1+x^2}{1+2^x}dx$$
First I tried something ...
- 147
6
votes
1
answer
353
views
Deriving the Integral for Alternating Harmonic Series Partial Sums
The partial sums of the harmonic series (the Harmonic Number, $H_n$) are given by
$$H_n=\sum_{k=1}^{n} \frac{1}{k}$$
and the well known integral representation is
$$H_n=\int_0^1 \frac{1-x^n}{1-x}\,dx$$...
- 3,667
10
votes
5
answers
631
views
Evaluate $\int_{0}^{\pi} \frac{x\coth x-1}{x^2}dx$
I've been trying to evaluate certain series recently, and I found that
$$\sum_{r=1}^{\infty}\frac{1}{r}\arctan\frac{1}{r}=\frac{\pi}{2}\int_{0}^{\pi} \frac{x\coth x-1}{x^2} \, dx$$
Therefore, I would ...
- 6,560
0
votes
1
answer
54
views
Compute integral without using derivative
I want to calculate the following integral without using derivatives
$$\frac{1}{b-a}\int_a^b e^{ty}\,{\rm d} y$$
where $t, a, b \in \Bbb R$. I know that the result is
My first idea was to transform ...
- 9
0
votes
1
answer
49
views
How to solve an integral inside a summation (with a divergent term)
I have a question that might be silly, but I really don't understand what is going on. I have to solve the following integral:
$$ \sum_{n \in \mathbb{Z}} \int_{m}^{m+1}e^{inx}dx $$
However if we try ...
- 91
1
vote
1
answer
170
views
Using Cauchy product for an integral
To evaluate
$$
\int_0^1 e^x \ln(x+1)dx
$$
I was thinking about using the cauchy product of the taylor series of $e^x$ and $\ln(x+1)$. We know that
$$
e^x = \sum_{n=0}^\infty \frac{x^n}{n!}
$$
and
$$
\...
5
votes
3
answers
294
views
Find the closed-form solution to a integral with the floor function
I have the following integral:
$$\int_0^1k^{\left\lfloor\frac{1}{x}\right\rfloor}dx$$
My question is, is there a nice closed form of this integral?
I have not any idea where to start.
Maybe I could ...
user794351