All Questions
17
questions with no upvoted or accepted answers
7
votes
0
answers
453
views
Can we interchange the Integral and Summation when a limit is $\infty$?
I was trying to Evaluate the Integral:
$$\Large{I=\int_1^{\infty} \frac{\ln x}{x^2+1} dx}$$
$$\color{#66f}{{\frac{1}{x^2+1} = \frac{1}{x^2\left(1+\frac{1}{x^2}\right)}=\frac{1}{x^2}\cdot \frac{1}{1+...
- 3,064
6
votes
0
answers
535
views
Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$
Hi I am trying to integrate and obtain a closed form result for
$$
I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx.
$$
Here is what I tried (but I do not think this is ...
- 9,966
4
votes
0
answers
135
views
Simplify a summation in the solution of $\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x$
Context
I calculated this integral:
$$\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
0
answers
226
views
integrability question, lower\upper integral and the lower\upper Riemann sum
Define a function $g : [0, 1] \mapsto \mathbb R$ by the following formula:
$$g(x)=\begin{cases}-1,&x\in \mathbb{Q}\\x^3-x,&x\not\in \mathbb{Q}\end{cases}$$
(a) What is $\underline{I}^...
- 5,283
3
votes
0
answers
253
views
Finding upper and lower sum of a Heaviside function.
I have been asked to find the lower and upper sum of a Heaviside function. It is a combination of a Heaviside function, like: H(x-1) + (3x-12)H(x+1).
I have been able to draw the graph for this, ...
- 89
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:
$${\int_{0}^{\frac{\pi}{2}}\ln\left(1+\alpha^n\sin(x)^{2n}\right)\mathrm{d}x=\frac{n\pi}{...
2
votes
0
answers
84
views
Is it possible to compute this integral using this double summation?
I attempted to evaluate the following integral, $$\int_{0}^{1} \frac{\ln (1+x)}{1+x^2}\, dx$$
although I know it can be evaluated using Feynman's Technique of Integration, Trigonometric Substitution, ...
- 423
2
votes
1
answer
107
views
Find the value of $\int_0^1{4dx\over 4x^2-8x+3}$
Find the value of $\displaystyle\int_0^1{4dx\over 4x^2-8x+3}$
$$\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
0
answers
107
views
Why are integrals and summations useful in computer science and what do these function mean?
I am reading my way through an introductory MIT computer science book called Structure and Interpretation of Computer Programming, and while I understand the programming and logic behind the book, ...
- 231
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
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^...
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
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
0
votes
2
answers
144
views
Approximate the integral $\int_0^{0.5}{x^2e^{x^2}}dx$ correct to four decimal places using a Maclaurin series.
I got $$\int_0^{0.5}{\sum_0^\infty}\frac{x^{2n+2}}{n!}dx$$ for the taylor series representation, but I'm not sure what to do next.
Do I use 0 and 0.5 as bounds for z for the Lagrange Error Bound? And ...
user738554
0
votes
0
answers
63
views
Find the limit when $n \rightarrow \infty$ of the series
Find the limit when $n \rightarrow \infty$ of the series:
$$\frac n{n^2}+\frac n{n^2+1^2}+ \frac n{n^2+2^2}+\cdots+\frac
1{n^2+(n+1)^2}$$
I am required to do this using limit of a sum definition ...
- 1,147