All Questions
239
questions
2
votes
1
answer
107
views
Find the value of $\int_0^1{4dx\over 4x^2-8x+3}$
Find the value of $\displaystyle\int_0^1{4dx\over 4x^2-8x+3}$
$$\begin{align*}\int_0^1{4dx\over 4x^2-8x+3}&=\int_0^1{dx\over (x-1)^2-(\frac 12)^2}
\\&=\int_0^1{dx\over (x)^2-(\frac 12)^2}
\\&...
- 3,396
0
votes
0
answers
86
views
Can I change the step interval on a summation from 1 to something else?
I have seen integrations written as summations like this,
$$\int f(x)\,dx=\lim_{\delta x\to0}\sum_{x=0}^x f(x)\,\delta x\tag{1}\label{one}$$
But I feel like that $\sum$ notation doesn't really explain ...
- 1
2
votes
3
answers
146
views
Computing $\int_{-2}^{2}\frac{1+x^2}{1+2^x} dx$
I am trying to compute the following integral by different methods, but I have not been able to come up with the result analytically.
$$\int_{-2}^{2}\frac{1+x^2}{1+2^x}dx$$
First I tried something ...
- 147
0
votes
2
answers
40
views
Why $\sum_{k=1}^{\infty}\int_{\frac{(4k+1)\pi}4}^{\frac{(4k+3)\pi}4}\frac{dx}x\ge\sum_{k=1}^{\infty}\frac{\pi}2\times\frac{4}{(4k+3)\pi}$?
This is a step of an answer that I don't completely understand:$$\sum_{k=1}^{\infty}\int_{\frac{(4k+1)\pi}4}^{\frac{(4k+3)\pi}4}\frac{dx}x\ge\sum_{k=1}^{\infty}\frac{\pi}2\times\frac{4}{(4k+3)\pi}$$
I ...
- 6,794
2
votes
1
answer
39
views
Questions involve Integrals and Sequence
Let $(a_n)_{n=1}^ \infty$ be a non-negative sequence, let $k ∈ \mathbb{N}$ and let $f : [k, ∞) → \mathbb{R}$
be a function that is integrable on $[k, b]$ for every $b > k$.
Prove or disprove each ...
- 627
1
vote
2
answers
201
views
Determine if $\int_1^\infty \arctan(e^{-x})dx$ converges or diverges
I want to check if the sequence $\sum_{n=1}^\infty [\sin(n) \cdot \arctan(e^{-n})]$ converges absolutely, by condition, or not.
We know that $\sum_{n=1}^\infty [\sin(n) \cdot \arctan(e^{-n})] \le \...
- 627
15
votes
2
answers
865
views
Evaluate $\sum\limits_{n=1}^{\infty}\frac{1}{n^3}\binom{2n}{n}^{-1}$. [duplicate]
Evaluate $$\sum\limits_{n=1}^{\infty}\frac{1}{n^3}\binom{2n}{n}^{-1}.$$
My work so far and background to the problem.
This question was inspired by the second page of this paper. The author of the ...
- 6,560
6
votes
1
answer
353
views
Deriving the Integral for Alternating Harmonic Series Partial Sums
The partial sums of the harmonic series (the Harmonic Number, $H_n$) are given by
$$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$$...
- 3,667
1
vote
0
answers
86
views
Series representation of $\int_{0}^{1}\frac{\sin(x^p)}{x}dx$
Consider the integral $\int_{0}^{1}\frac{\sin(x^p)}{x}dx$ for $p>0$. Assume that $\sum_{n=0}^{\infty}a_n$ is the series representation of the integral.
Then, does $\sum_{n=0}^{\infty}a_n$ ...
- 85
0
votes
0
answers
156
views
Formula for partial fractions with repeated roots
I've been trying to find a formula for the partial fraction of $\frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials, $\operatorname{deg}(P(x))<\operatorname{deg}(Q(x))$ and $Q(x)$ has ...
- 436
10
votes
5
answers
631
views
Evaluate $\int_{0}^{\pi} \frac{x\coth x-1}{x^2}dx$
I've been trying to evaluate certain series recently, and I found that
$$\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$$
where $t, a, b \in \Bbb R$. I know that the result is
My first idea was to transform ...
- 9
3
votes
3
answers
205
views
Integral of the inverse of a "simple" sum
$$\Large \int_{0}^{\infty} \left (\frac{1}{ \sum_{k=0}^{\infty} x^k} \right)~ dx$$
It seems like this converges to $\approx \frac{1}{2}$ but when I try to calculate this I get really stuck, and for no ...
0
votes
0
answers
98
views
How would I evaluate this limit using summation equations?
This is an alternate way of finding the area under a curve for a certain function:
$$\lim_{n \to \infty} \sum_{k=1}^n \frac{3}{n} \sqrt{9-\left(\frac{3k}{n}\right)^2}$$
This expression would give:
$$\...
- 35
0
votes
0
answers
102
views
Definite integrals as Riemann sums
The Riemann sum/integral, is defined to be
$$ \int_a^b f(x)dx := \lim_{n,\Delta x_i \rightarrow 0} \sum_{i=1}^n f(x_i^*)\Delta x_i $$
whenever the sum exists, where $\Delta x_i$ is the sub-interval ...
- 852