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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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1answer
14 views

How to determine the radius when utilizing the disc method

Lately, I have been struggling to figure out how to determine the radius used in the formula $V=\pi r^2$ when finding the volume of a solid revolving around an axis. From my experience with solving ...
1
vote
1answer
28 views

Integral with log of absolute value of sine

Show that $$\int_{-\pi/3}^{\pi/3} \log \vert 8 \sin(t/2) (1 + \sin t)^2 \vert dt = 0.$$ WolframAlpha claims that this is true. I've tried manipulating the integrand a bunch and various trig ...
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2answers
59 views

Suppose that the average value on all intervals $[a,b]$ is equal to $f((a+b)/2)$. Prove that $f''(x) = 0$ for all $x \in \mathbb{R}$

I understand that $f(x)$ must be linear with a first derivative equal to a constant. I'm just not sure how I can use the mean value property of integrals to show something about $f''(x)$. The hint on ...
1
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0answers
64 views

Tough integral from particle physics

I have been struggling to check if it's even possible to calculate the following integral $$ \iiint_{\Bbb R^3} d^3 \textbf{q} {1\over \sqrt{E_{\textbf{p}+\textbf{q}}(E_{\textbf{p}+\textbf{q}}+m)}}e^{-...
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2answers
45 views

Proving Integrals

How to prove that $$ \int_{0}^{\frac{\pi}{2}} \sqrt{\tan x} \ dx = \int_{0}^{\frac{\pi}{2}} \sqrt{\cot x} \ dx$$ I have no idea what I'm gonna do first. I just checked the graph, and the result is the ...
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0answers
23 views

How to derive the closed form for $\int_0^1 \left(\ln(\Gamma(z))\right)^2dz$

Is there any way to derive this identity?$$\int_0^1 \left(\ln(\Gamma(z))\right)^2dz=\frac{\gamma^2}{12}+\frac{\pi^2}{48}+\frac{\gamma\ln(2\pi)}{6}+\frac{(\ln(2\pi))^2}{3}-(\gamma+\ln(2\pi))\frac{\zeta'...
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0answers
24 views

Number of solutions within an interval

What I'm trying to calculate is the following $$\int_0^{2\pi}dz_1\sum_{j\text{ s.t. } z_2^{(j)}=\arccos(\lambda-\cos(z1))\text{ or }\\\text{ written another way}\\0\leq\arccos(\lambda-\cos(z1))\leq\...
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1answer
43 views

Prove $\sqrt{1-e^{-1}}<\frac{1}{\sqrt{\pi}}\int^1_0 e^{-x^2}dx<\sqrt{1-e^{-2}}.$

Problem Prove $$\sqrt{1-e^{-1}}<\frac{1}{\sqrt{\pi}}\int^1_0 e^{-x^2}dx<\sqrt{1-e^{-2}}.$$ Attempt Since $$ \forall x\in[0,1],\quad e^{-x^2}=\sum_{n\geq 0}\frac{(-1)^n}{n!}\,x^{2n}$$ and by ...
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1answer
37 views

Is there a way to mathematically prove $\psi (\mathbf{r})$ varies continuously (using the intuitive arguments provided below)?

Electric potential at a point outside the charge distribution is: $\displaystyle \psi (\mathbf{r})= \int_{V'} \dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'$ where: $\mathbf{r}=(x,y,z)$ ...
4
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1answer
108 views

Integrals of the form $\int \log(2+2\cos x)^ndx$

$\log$ will be the natural logarithm and $\zeta$ the Riemann zeta function. I'm interested in the following family of integrals: $$ I_n = \int_0^\pi(\log(2+2\cos x))^n\mathrm{d}x $$ Some of the values ...
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3answers
54 views

If $I_{n}=\int^{1}_{0}x^2(1-x^2)^ndx,$ Then $\lim_{n\rightarrow \infty}\frac{I_{n+1}}{I_{n}}$ is

If $\displaystyle I_{n}=\int^{1}_{0}x^2(1-x^2)^ndx,$ Then $\displaystyle \lim_{n\rightarrow \infty}\frac{I_{n+1}}{I_{n}}$ is plan $$I_{n}=-x\cdot \frac{(1-x^2)^{n+1}}{2(n+1)}\bigg|^{1}_{0}+\frac{...
3
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2answers
59 views

Remove even elements of partition of integration set

Suppose I am integrating a continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ in a measurable set $A\subseteq I$, where $I$ is an interval: $$ \int_{A}f(x)dx $$ Now suppose I partition the set $I$ in $N$ ...
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0answers
54 views

Tricky integral limit [duplicate]

Please help me to solve the following limit using high-school methods(no approximations or DCT). I was thinking to use the squeeze theorem but I can't find the best boundaries to evaluate it. $$\lim_{...
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1answer
40 views

How was this integral with functions as limits solved? [duplicate]

How was this integral solved ? $$\displaystyle \frac d{dy}\int_{g(y)}^{h(y)}f(x,y)dx$$ $$=\int_{g(y)}^{h(y)}\frac \partial{\partial y} f(x,y)dx+f(h(y),y)\frac{dh(y)}{dy}-f(g(y),y)\frac{dg(y)}{dy}$$ ...
1
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1answer
69 views

Evaluation of integral $\int_{-\infty}^{\infty} e^{itx} \frac{1- e^{-\frac 1 2 t^2}}{\frac 1 2 t^2} \text d t$ / specific characteristic Function

I want to calculate the value of $$I(x) :=\int_{-\infty}^{\infty} e^{itx} \frac{1- e^{-\frac 1 2 t^2}}{\frac 1 2 t^2} \text d t$$ where $x\in \Bbb R$. Of course we can write $$I(x) = \int_{-\infty}^{\...
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2answers
61 views

Even stronger than Sophomore's dream [duplicate]

Sophomore's dream states that: $$ \int_0^1x^{-x}dx=\sum_{n=1}^\infty n^{-n} $$ and $$ \int_0^1x^{x}dx=-\sum_{n=1}^\infty(-n)^{-n} $$ A friend of mine noticed that numerically: $$ \int_0^1\int_0^1(xy)^{...
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2answers
30 views

How to explain or give reasons about the curve?

I know that arc length of the curve is: $$s=\int_{a}^{b}\sqrt{1+\big(f^{'}(x)\big)^2}dx$$ The question is "Is there a smooth curve y=f(x)" where length over the interval $0\leq x \leq a$, (where a is ...
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1answer
32 views

How to find the length of the curve?

I know that length of the curve is either of: $$ s = \int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}\ dx = \int_c^d \sqrt{1+\left(\frac{dx}{dy}\right)^2}\ dy$$ Now the curve is $y=x^2$, $-1\leq x\...
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2answers
43 views

integration of a gaussian with $x^2$

I need to integrate $$\int_{-\infty}^{\infty} x^2 e^{-ax^2} \qquad \text{where } a\in R$$ The book does the following: I don't understand what's happening. I tried solving the integral using ...
1
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0answers
76 views

Closed-forms for the integral $\int_0^1\frac{\rm{Li}_n(x)}{1+x}dx$?

(This is related to this question.) Define the integral, $$I_n = \int_0^1\frac{\rm{Li}_n(x)}{1+x}dx$$ with polylogarithm $\rm{Li}_n(x)$. Given the Nielsen generalized polylogarithm $S_{n,p}(z)$, $$...
5
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0answers
137 views

More on the integral $\int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$

In this post, the OP asks about the integral, $$I = \int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$$ I. User DavidH gave a beautiful (albeit long)...
2
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2answers
112 views

Hard and weird Integration in numerator and denominator

$$ \frac{\int_0^\pi x^3\ln(\sin(x))\ dx} {\int_0^\pi x^2\ln(\sqrt{2}(\sin(x))\ dx}$$ In this problem , I'm unable to understand how to start. I tried applying By parts but I couldn't solve it . ...
4
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0answers
33 views

Convolution of a box function over the hypercube

This looks like a simple convolution over the $p$-dimensional hypercube, but I am unable to find a closed form expression for arbitrary integer $p$: $$f_p(a)=\int_0^1 dx_1\int_0^1 dx_2\cdots \int_0^1 ...
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0answers
38 views

Evaluate the definite integral $\int_0^{\infty} \frac{x}{e^x-e^{-x}}$ [duplicate]

Evaluate the definite integral $$\int_0^{\infty} \frac{x}{e^x-e^{-x}}$$ I couldn't get the system at first that is why I deleted my first post. I couldn't find how to edit it so I decided to ask here....
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1answer
69 views

Find the value of $\int_0^1xf(x)dx$

Given: $f(0)=0, f(1)=2, \text{ and} \int_0^1f(x)dx=3$ Find the value of $\int_0^1xf(x)dx$ Attempt: Using partial integration. $$ \int xf(x)dx=\frac{x^2f(x)}{2}-\int\frac{x^2f'(x)}{2}dx $$ Maybe I'...
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2answers
41 views

Prove $|\int_{n}^{n+p} \sin (x^2)dx|\leq 1/n$ where $p>0$.

Prove $$\left|\int_{n}^{n+p} \sin (x^2)dx\right|\leq \frac{1}{n}$$ where $p>0$. Maybe, we can improve namely enhance to prove $$\left|\int_{n}^{+\infty} \sin (x^2)dx\right|\leq \frac{1}{n},$$ ...
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0answers
34 views

Finding Limit involving definite integral [on hold]

Suppose the limit $$L = \lim_{n\to\infty} \sqrt{n} \int_{0}^1 \frac{1}{(1+x^2)^n}dx$$ exists and is larger than $\frac{1}{2}$. Then which of the following inequalities are true? A) $...
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1answer
27 views

Evaluating $\int_{-\pi}^{\pi} dq \; e^{iqn} e^{i \alpha\cos(q)}$

I am trying to solve this definite integral $$\int_{-\pi}^{\pi} dq \; e^{iqn} e^{i \alpha\cos(q)}$$ where $\alpha \in\mathbb{R} $ and $n\in\mathbb{Z}$. I know that for $n=0$ $$\int_{-\pi}^...
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0answers
19 views

How to set up a line integral of a vector field over the borders of two circumferences

So...I have found a problem where I have to solve the integral of the vector field: $F(x,y)=(\sin(x)ln(x)+y^2) a_x + (\cos(y)e^y-x^2) a_y$ Along the borders of the region bounded by the ...
2
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2answers
54 views

Is $\int_0^1 \frac{x^n}{\sqrt{1-x^4}}dx$ convergent?

$$\int_0^1 \frac{x^n}{\sqrt{1-x^4}}dx$$ Near $0$ the expression inside is convergent, that is easy. Near $1$ looks like it approaches infinity when $n \ge 0$ But according to the book when $n \ge -1$ ...
1
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1answer
70 views

How can I solve this kind of integral? [on hold]

Evaluate $$\int_{-\infty}^{\infty}\frac{1}{z}e^{-i(Az^2 - Bz)}dz,$$ where A and B are some coefficients.
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2answers
32 views

Prove $\ln \int_0^1 f(x)dx \geq \int_0^1 \ln f(x) dx$.

Let $f(x) \in C[0,1]$, and $f(x)>0$ over $[0,1]$. Prove $$\ln \int_0^1 f(x)dx \geq \int_0^1 \ln f(x) dx.$$ If we denote $$F(x):=\ln \int_0^x f(t){\rm d}t-\int_0^x \ln f(t){\rm d}t, ~~~x \in[0,1]$$ ...
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2answers
61 views

Let $f:[a,\infty)\rightarrow \mathbb{R}$ be a uniformly continuous function. $\int_{a}^{\infty} f$ converges.Prove that $\lim_{x\to\infty} f(x)=0$

Let $f:[a,\infty)\rightarrow \mathbb{R}$ be a uniformly continuous function in that range. $\int_{a}^{\infty} f$ converges. Prove that $\lim_{x\to\infty} f(x)=0$ Hint: Use the sequence $F_n(x)=n\...
1
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1answer
48 views

Differentiation when there are continuously many variables

Suppose there are a continuum of tasks in a unit range $[0, 1]$, and for each task $i \in [0, 1]$, a firm can choose the amount of robots, $x_i$. I am hoping to get the first-order condition (e.g., $\...
3
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0answers
50 views

Can $1/\log(2)$ be represented as a period?

In this article by Zagier-Kontsevich, period is defined as values of integral of a rational function over a domain in $\mathbb{R}^{n}$ defined by polynomial inequalities with rational coefficients. ...
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0answers
59 views

Evaluate $\int_0^1 \frac{\ln(1-x^2)\arcsin^2 x}{x^2} {\rm d}x$ [duplicate]

Evaluate $$I=\int_0^1 \frac{\ln(1-x^2)\arcsin^2 x}{x^2} {\rm d}x.$$ Maybe, we can make a substitution $x=:\sin u, u\in [0,\pi/2]$. Then $$I=2\int_0^{{\pi}/2}\frac{u^2\cos u\ln\cos u}{\sin^2 u}{\rm d}...
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3answers
100 views

Can't solve $\int_{0}^{\pi} \frac{x}{1 + \cos^2x} dx$

I tried this :- Let $$I =\int_{0}^{\pi}\frac{x}{1 + \cos^2x}dx\tag{1}$$ then $$I = \int_{0}^{\pi}\frac{\pi-x}{1 + \cos^2(\pi-x)}dx= \int_{0}^{\pi}\frac{\pi-x}{1 + \cos^2x}dx\tag{2}$$ Adding (1) and (...
0
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3answers
39 views

two dimensional integral of delta function

For $x,y \in \mathbb{R}$, function $f(x,y)$ is defined as $$f(x,y) = 1 \quad\textrm{if}\quad x=y$$ $$f(x,y) = 0 \quad\textrm{if}\quad x\neq y$$ It seems to me that the integral $I = \int_0^1 \int_0^1 ...
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votes
3answers
85 views

Problem with integrating $\int_0^{\pi/2}\frac{\cos^6x}{\cos^6x+\sin^6x}dx$ [duplicate]

Someone told me there is an equation $$\int_0^{\pi/2}f(sinx)dx=\int_0^{\pi/2}f(cosx)dx$$ With this equation, it's easy to get the answer$\frac{\pi}{4}$. What I want to know is why we have this ...
0
votes
1answer
28 views

Help with Contour Integration with Finite Limits

I need to perform this integral: $$ \int_0^{\omega_{\large s}/\left(4\pi\right)} \frac{\mathrm{i}\omega\,\mathrm{e}^{\mathrm{i}\omega t}} {\left(\mathrm{i}\omega - \omega_{0}\right) \left(\mathrm{i}\...
0
votes
1answer
33 views

Proving the accuracy for numerical integration

Given a smooth function $f$, we denote $L_{f}$ the Lagrange polynomial of degree less than or equal to $1$ which is equal to $f$ at the points $x_1$ and $x_2$. Define $I_{f} = \int_{-1}^{1} L_{f}(...
1
vote
0answers
35 views

What are the similarities between these 3 limits?

I have 3 limits and I'm pretty sure that the process of solving them it's the same. $L_1=\lim_{n\rightarrow \infty }n\cdot \int_{0}^{1}x^{2n}\sin(\frac{x\cdot \pi)}{2})dx$ $L_2=\lim_{n\rightarrow \...
0
votes
1answer
35 views

$\lim_{n \to \infty} n \int_{0}^{100}f(x)g(nx)dx=f(0)$

Let $g: \mathbb{R} \to \mathbb{R}$ be a continuous function with $g(y)=0$ for all $y \notin [0,1]$ and $\int_{0}^{1}g(y)dy=1$. Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function. ...
1
vote
2answers
56 views

Can a certain series of integrals over $[0,\frac{1}{16}]$ be solved using integration-by-parts?

I have a series of six integrals. The first, say, is \begin{equation} \int_ 0^{\frac {1} {16}} - 48 q^{3/2}\sqrt {18 q - 2\sqrt {17 q + 4}\sqrt {q} + 4} \mbox {d} q. \end{equation} Mathematica ...
0
votes
2answers
50 views

how can i find $\lim _ { n \rightarrow \infty } \int _ { 0 } ^ { \pi / 2 } e ^ { - n \sin x } d x$

i need to find $\lim _ { n \rightarrow \infty } \int _ { 0 } ^ { \pi / 2 } e ^ { - n \sin x } d x$ What i tried: $$\left|\int _ { 0 } ^ { \pi / 2 } e ^ { - n \sin x }\mathrm{d}x\right|\leq\int _ { 0 } ...
2
votes
3answers
42 views

Evaluate $\int^{2}_{0}f(x)dx$

Let $f(x)=f(-x)$ for $x\in \mathbb{R}$. If $\int^{3}_{-3}f(x)dx=0$ and $\int^{3}_{-2}f(x)dx=5$, then $\int^{2}_{0}f(x)dx=...$ First I tried this: $\begin{aligned} \int^{3}_{-3}f(x)dx&=2\int^{3}_{...
1
vote
2answers
68 views

How do I do this integral $\int_{-\pi}^{\pi}\frac{\cos^2(x)}{1+a^x}dx, a>0$?

$$\int_{-\pi}^{\pi}\frac{\cos^2(x)}{1+a^x}dx, a>0$$ I've tried doing $z = e^{ix}$ and evaluating the resulting contour integral, but this introduced a branch cut that goes through the contour ($a^x$...
8
votes
2answers
171 views
+50

Evaluating sum $\sum_{m=0}^{\infty}\frac{(2-\delta_m^0)(-1)^m \lambda_0}{a(\lambda_0^2 -(\frac{m\pi}{a}))}\cos(m\pi x/a)$

How Can I evaluate the following sum$$\sum_{m=0}^{\infty}\frac{2-\delta_m^0}{a}\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})}\cos\left(\frac{m\pi x}{a}\right)=\frac{\cos(\lambda_0 x)}{\sin(\...
-3
votes
0answers
37 views

Solving double integral [closed]

Can someone help me to solve the following double integral; where $a_1,\epsilon_1 ,\Phi, p, a_2 ,k$ are constants and $a_2 -a_1 = \epsilon_1$; integral to solve
1
vote
1answer
57 views

Solve the numerical value of this integral $\int_0^\infty t^{a-1}e^{-t} \Gamma(b,t)dt$

I need to compute the numerical value of this integral, hundred thousand of times, for a typical dataset. How can I get a good approximation. $$ \int_0^\infty t^{a-1}e^{-t} \Gamma(b,t)dt $$ where a ...