Questions about the evaluation of specific definite integrals.

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17 views

Asymptotics of integrals

I am struggling with this problem where we're asked to use method of steepest descent: Find the leading term of the asymptotics of the following integral for $\lambda\to\infty$ : ...
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1answer
22 views

Convert Riemann sum to definite integral $ \lim_{n\to\infty}\sum_{i=1}^{n} \frac{21 \cdot \frac{3 i}{n} + 18}{n} $

The limit $\quad\quad \displaystyle \lim_{n\to\infty}\sum_{i=1}^{n} \frac{21 \cdot \frac{3 i}{n} + 18}{n} $ is the limit of a Riemann sum for a certain definite integral $\quad\quad \displaystyle ...
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2answers
19 views

Compute variance of logistic distribution

Consider a random variable $X$ with normalized logistic distribution( so that its pdf is $\frac{e^{-x}}{(1+e^{-x})^2}$). It is well known that its variance $V$ equals $\frac{\pi^2}{3}$ but I couldn't ...
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0answers
21 views

Triple integral boundaries

If $W\subset \mathbb{R}^3$ is bounded by the planes $y+z=2, 2x=y, x=0, z=0$, what are the boundaries of $\int\int\int_W x dV$? How can I find the boundaries if I take $dV$ as $dydxdz$, $dxdzdy$ and ...
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2answers
18 views

Analytic geometry and definite integrals problem…

So, here's the problem: We have a parabola $y^2=2px$ and a line which is perpendicular to parabola and forms the angle $\frac{3\pi}{4}$ with x axis. I have to find the area between the parabola and ...
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0answers
36 views

How do I find the volume of this torus?

A torus $T_{a,b}\subset \mathbb{R}^3$ is described with torus coordinates $$x=(a+\rho \cos\theta)\cos\phi,\space y=(a+\rho \cos\theta)\sin\phi,\space z=\rho\sin\phi,$$ with $a>0$. Now $T_{a,b}$ is ...
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1answer
28 views

Closed-form expression for $\int_{0}^{1}e^{-ax(1 - bx )}x^{\alpha-1}(1-x)^{\beta - 1}dx$?

As per the title, I am looking for a closed-form expression for the integral $$\frac{1}{B(\alpha,\beta)}\int_{0}^{1}e^{-ax(1 - bx )}x^{\alpha-1}(1-x)^{\beta - 1}dx$$ where $a,\alpha,\beta>0$ and ...
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2answers
49 views

I need a Hint on this Differential equation

Let $f$ and $g$ be two real functions, and we have $$ (f*g)(t)= \int_0^tf(s)g(t-s) \, ds $$ we have the following equation $$y'+ay=f(t) $$ where a is a constant and f is a function -/ prove that if ...
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1answer
24 views

Evaluating area using an integral in polar coordinates

I am trying to find the area of a circle which is given by the polar parameterization $$r(\phi) = \cos\phi + \sin\phi.$$ I can evaluate it in 2 ways and don't know why I get different answers. First ...
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2answers
55 views

A problem in definite integral.

What will be the value of $a$ for which the integral $$\int \limits^{\infty }_{0}\frac{dx}{a^{2}+(x-\frac{1}{x})^{2}} =\frac{\pi}{5050}$$ where $a^{2}\geq0$ It seems like a standard integral but ...
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1answer
33 views

finding the value of $a$ for minimal solid

Given that $y=x^2+ax-0.25$ is bounded by $x=0$ and $y=1$ and spinning around the x-axis. What is the value of $a$ for min volume of that solid? My attempt: $x^2+ax-0.25=1$ so $x=\frac{-a\pm ...
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3answers
66 views

Computing $\int_{0}^{\infty} \frac{x^2 + a}{x^6 + a^3}dx$

I wish to compute $$\int_{0}^{\infty} \frac{x^2 + a}{x^6 + a^3}dx, \quad a>0$$ but have no contour to work with. Does anyone have ideas on how to compute this integral?
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2answers
211 views

Basic integration question.

I have the integral $$\iint x^2y^2 \ dx\,dy$$ but I am meant to evaluate it at the limits $0<y<1$ and $-2y<x<2y$. I am wondering what terminals of integration I should put in for $x$. Do I ...
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2answers
72 views

What methods to use to integrate $\sqrt{1+t^4}$?

I have this integral to evaluate $$\int^x_1 \sqrt{1+ t^4}\, dt$$ I have tried substitution, trig identity and integration by parts, i don't have any answer. Can anyone explain the method I need to ...
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1answer
24 views

Combinations of exponentials and arbitrary powers integration [on hold]

I don't know how to solve this equation: $$\int_0^\infty e^{-x} (x-a)^n dx$$ where $a$ is a constant. Thanks in advance for your help.
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3answers
47 views

Simple Integration by Substitution requiring bizarre answer

Before addressing my queries and attemps I shall be posting the full question below. Use the substitution $x=e^u$ to find $$\int (\ln x)^2dx$$ My answer boiled down to $\dfrac{2x^3}{3} + C$ ...
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2answers
35 views

Double integrals- cartesian to polar [on hold]

$$\int^\infty_{-\infty}\int^\infty_{-\infty} \frac{1}{a^2 + x^2 +y^2}\,dy\,dx$$ How can I convert the integral to polar form the hint given in the question is:x-y plane
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0answers
29 views

Proving if $f(x)$ is an integrable function on $[a,b]$ then $g(x)=f(x-c)$ is integrable on $[a+c,b+c]$

Prove that if $f(x)$ is an integrable function on $[a,b]$ then $g(x)=f(x-c)$ is integrable on $[a+c,b+c]$. My attempt: Since $f$ is integrable then there's a sequence of partitions ...
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2answers
41 views

Limits and Integration Problem

I have no idea as to how to go about this. Could somebody please help? Let $$\displaystyle ...
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25 views

Integrating $\int_0^1\cos(\lambda x^3)dx$ using the saddle point method [on hold]

Find the leading term of asymptotics as $\lambda\to\infty$ $I(\lambda)=\int_0^1\cos(\lambda x^3)dx$ Using method of saddle points along a certain contour. I am having trouble approaching this ...
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0answers
39 views

How can i evaluate the following limit?

I need to show that the following limit does converge to less than one depending on the values of $f(k)$ and $g(k)$: $$ \lim\limits_{k \rightarrow \infty} \frac{\frac{1}{f(k+1)} ...
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Definite Integration of a multivariable function [on hold]

$$ \int _{-\infty }^{\infty }\!\int _{ -\infty}^{\infty }\!\int _{ -\infty }^{\infty }\!\int _{ -\infty }^{\infty }\! 4.0\, \left( 0.06003683241\,{\frac {{\it k2}\,\sin \left( 10^{-8}\,{ \it k1} ...
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1answer
8 views

The region bounded by the curve

What is the the region bounded by the curve y=sin x, and the line y=1, and the y-axis to the right of the y-axis?
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2answers
31 views

Area of region bounded by curves. [on hold]

What is the region bounded by the curve $y=\tan^2 x$, the $x$-axis and the line $x=\frac{1}{4}π$.
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29 views

Can someone explain how to solve these 2 mathematics questions? [on hold]

Question 1: The quadratic function which takes the value $41$ at $x=-2$ and the value $20$ at $x=5$ and is minimized at $x=2$ is $$y=Ax^2-Bx+C$$ The minimum value of this function is $D$. Find ...
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4answers
170 views

Calculate in closed form $\int_{0}^{1}\int_{0}^{1}\frac{dxdy}{1-xy(1-x)(1-y)}$

Can we possibly compute the following integral in terms of known constants? $$\displaystyle \int_{0}^{1}\int_{0}^{1}\frac{dxdy}{1-xy(1-x)(1-y)}$$ Some progress was already done here ...
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1answer
40 views

Convergence of series with integral test

Given that the following series is convergent, determine the values of p. $$\sum_{n=2}^{\infty}\dfrac{1}{n(\log(n))^p}$$ So far what I have done is using the integral test, in order to use integral ...
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1answer
53 views

Can anyone help with these improper integrals? [on hold]

$$\iiint_D{{e^{\sin(x+y+z)} \over x^2+y^2+z^2}dxdydz} ;D=\{(x,y,z),x^2+y^2+z^2\geq 1\}$$ For this one and the next it seems like spherical coordinates should be applied and in this case the $D_n={1 ...
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0answers
27 views

Laplace transform of Gaussian*Erfi

$$\sqrt{\frac{\pi }{2}} e^{-\frac{t^2}{2}} \text{erfi}\left(\frac{t}{\sqrt{2}}\right) \rightarrow -\frac{1}{2} e^{\frac{s^2}{2}} \text{Ei}\left(-\frac{s^2}{2}\right)$$ or $$ ...
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1answer
41 views

Understand Substitution used in Integral

This is a solution I had come across for a general case.$$$$ We will use $$\Gamma(x)=\int_0^{\infty}t^{x-1}e^{-t}\,dt$$ $$\Gamma(\frac{1}{2}+n)=\frac{(2n-1)!!}{2^n}\sqrt{\pi}$$ & we will first ...
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35 views

How to convert a sum into a definite integral?

I know there is some way to convert a sum into an integral, and vice versa. However, I am very confused on how actually to do this, and if there is some sort of intuition behind it. Note: I'm not ...
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1answer
22 views

Proof that the distibution sequence $\int_{0}^{\infty}(tanh(nx)-1)\phi(x)dx \to 0$

Let $T:\phi \mapsto \langle T_f,\phi\rangle =\int _{-\infty}^{\infty}f(x)\phi(x)dx$, where $\phi(x)$ is a test function, be a distribution. I would like to prove that the sequence of distributions: ...
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30 views

Prove that for every $n \in \mathbb N$, $\int_{-\pi}^{\pi}f(x)\sin(2nx)dx=0$ if $f(x)$ is odd.

If $f:\mathbb R \to \mathbb R $ is odd continuous function such that $g(x):=f(x + \frac{\pi}{2})$ is even, prove that for every $n \in \mathbb N$, $\int_{-\pi}^{\pi}f(x)\sin(2nx)dx=0$. Since ...
3
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1answer
55 views

Method of Steepest descents integral

I am looking to evaluate the following asymptotic integral: Find the leading term of asymptotics as $\lambda\to\infty$ $I(\lambda)=\int_0^1\cos(\lambda x^3)dx$ Using method of steepest descents ...
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1answer
36 views

definite integral- convergence of a integral [on hold]

Fot which values of $p$ this integral converges? $$\int_e^\infty x^p(\ln x)dx $$ Do I have to make it equal to $0$?
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2answers
40 views

Orthogonality lemma sine and cosine

I want to know how much is the integral $\int_{0}^{L}\sin(nx)\cos(mx)dx$ when $m=n$ and in the case when $m\neq n$. I know the orthogonality lemma for the other cases, but not for this one.
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2answers
30 views

Evaluating $\int_{-5}^5\int_{-5}^5-\frac{3}{2}|x+y|-\frac{3}{2}|x-y|+15\,\mathrm{d}x\,\mathrm{d}y$

I'm always having the wrong result from the following: $$ \int_{-5}^5\int_{-5}^5-\frac{3}{2}|x+y|-\frac{3}{2}|x-y|+15\,\mathrm{d}x\,\mathrm{d}y $$ I would really appreciate some guidance on how to go ...
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1answer
37 views

How to calculate the integral $\int_0^2 { \int_0^{1/2x_1} {\frac{-1+x_1x_2-2x_2}{x_1-2x_2}} }dx_2dx_1$

This problem (if my derivations of them are correct) lead me to calculate the following integrals: $$I_1 = \int_0^2 { \int_0^{\frac{1}{2}x_1} {\frac{-1+x_1x_2-2x_2}{x_1-2x_2}} }dx_2dx_1$$ $$I_2 = ...
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1answer
42 views

Finding the definite integral using two variables - what am I doing wrong here?

I'm trying to find the average value of the function: $$p(t) = t7*sin0.2t^2+75 \quad dt \quad on[0,12]$$ So I wanted to start off by first finding the definite integral. I'm being thrown off by the ...
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1answer
39 views

Can someone please show step-wise how to integrate the following?

Supposedly, integration and simplification of $$a_n=\frac{2}{l} \int_0^a\frac{hx}{a}\sin \frac{n\pi x}{l}dx + \frac{2}{l}\int_a^l \frac{h(l-x)}{l-a} \sin \frac{n \pi x}{l} dx$$ is to yield ...
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1answer
43 views

Prove that $\frac{du}{dt}=\delta(t)$ where $u(t)$ is Unit step function, $\delta(t)$ is impulse function

I want to prove that $$\frac{du}{dt}=\delta(t)$$ To do that I want to use the following: $$\int_{-\infty}^{\infty}\phi(t) * g^{(n)}(t)dt = (-1)^n\int_{-\infty}^{\infty}\phi^{(n)}(t)*g(t)dt$$ What I ...
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3answers
54 views

Area between $ 2 y = 4 \sqrt{x}$, $y = 4$, and $2 y + 4 x = 8 $

Sketch the region enclosed by the curves given below. Then find the area of the region. $ 2 y = 4 \sqrt{x}$, $y = 4$, and $2 y + 4 x = 8 $ Attempt at solution: I guess I'm supposed to divide the ...
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1answer
21 views

Show that $\lim_{r\uparrow 1}\frac{1}{2\pi}\int_{-\pi}^\pi f(x-y)\sum_{n=-\infty}^\infty r^{|n|}e^{iny}\,\mathrm{d}y=f(x)$ for any $2\pi$-periodic $f$

Let $0<r<1$ and consider the series $$s = \sum_{n=-\infty}^\infty r^{|n|}e^{inx}.$$ I have already shown that this series converges (uniformly) to $$P_r(x)=\frac{1-r^2}{1-2r\cos x+r^2}$$ ...
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0answers
16 views

Find multiple integral equation solution

$\int_{\theta=0}^{\pi}\int_{\phi=0}^{2\pi}((v\cos\theta\sin\phi+v')^2+(v\sin\theta\sin\phi)^2+(v\cos\phi)^2)\rho r^2\sin\theta d\theta d\phi$ Can you solve this equation please I use symbolab but I ...
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0answers
44 views

Area of two polar regions

I'm trying to find the region inside r=sinθ and outside r=1+cosθ. My issue is my limits of integration. I get an intersection at $\frac π2$ and one at the pole. What are my limits for the integral? ...
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1answer
145 views

Is $\int_{\mathbb R} f(\sum_{k=1}^n\frac{1}{x-x_k})dx$ independent of $x_k$'s for certain $f$?

My question below arises from the linked question $ \int\limits_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \leq n^{3/2}\pi$ and the comments by jack and David Speyer under that ...
2
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0answers
113 views

Confused about trying to find the correct spherical co-ordinates for this tricky triple integral

I'm having trouble trying to figure out how to change the limits of integration to spherical co-ordinates in this particular question. I was wondering if someone would kindly be able to assist me in ...
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1answer
22 views

Integral setup, vector calculus

The trajectory of an airplane is given by the parabola y = −x(x − 10)/10 where x and y are measured in km. Set-up an integral (with bounds) to calculate the amount of fuel burned by this airplane on a ...
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1answer
34 views

Why am I getting zero? Center of mass integral

Given a lamina $x^2 +y^2 = 1$ with density function $\sigma(x,y) = x+y$. Find the center of mass of the surface. So first lets find the mass. $$m= ...
15
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0answers
157 views

Proof of $\zeta(2)=\frac{\pi^2}{6}$

While messing around with some integrals, I have found the following proof for $\zeta(2)=\frac{\pi^2}{6}$, but I'm not sure if it is valid: We take a look at the integral $I=\int_0^{\frac{\pi}{2}} ...