Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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0
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1answer
50 views

Integration of step functions

I've managed parts (a) and (b) fairly easily, but c is causing me a real headache. I've seen the Cauchy-Schwarz inequality before, but I've hit a roadblock because I've no idea whether or not I can ...
-4
votes
0answers
17 views

the limits of integration for the following iterated integral [closed]

https://webwork-tr.ncc.metu.edu.tr/wwtmp/math120//gif/e210283-3264-setSummer2015Set3prob4image1.png i want to know how to solve this , i just care about the way to solve it
-1
votes
0answers
14 views

b spline, interpolation how many knots required? [on hold]

Hi I would like to get help with these questions. How many control points $d_i$ are involved when evaluating a cubic B-spline at a single points. The point are deboor. How many knots are necassary ...
2
votes
1answer
38 views

Double integral - Convert to polar coordinates and find the integration limits by a given domain [on hold]

I need help converting to polar coordinates and find the limits of the integrals by this given domain: $$\iint_{D}{} f(x,y)\, dx\, dy$$ $$D= \left\{ (x,y) \mid \dfrac {x^2}{a} \leq y\leq a, -a\leq 0 ...
8
votes
0answers
164 views

Help with the integral $\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$

Referring to a previous question, i want help with the integral : $$\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$$ Where $n,m$ ...
1
vote
2answers
36 views

partial fraction derivative question

So I have this partial fraction derivative question. I know how to solve it, but for some reason I keep swapping two numbers. Here is the problem: $$\int\frac{3-4x}{x^2+x}= ...
10
votes
0answers
170 views
+50

Integral formula for $\int_{0}^{\infty}e^{-3\pi x^{2}}((\sinh \pi x)/(\sinh 3\pi x))\,dx$ by Ramanujan

Towards the end of G. N. Watson's (one of the joint authors of famous book "A Course of Modern Analysis") paper "The Final Problem: An Account of the Mock Theta Functions" the following formula of ...
0
votes
1answer
25 views

Integrating a surface bound by a circle

I'm having an issue setting up this problem correctly, regardless of how I seem to do it I end up canceling everything out and getting $0$, which isn't the correct answer. $$ \text{Find the surface ...
1
vote
2answers
20 views

Determine region of integration for homework problem

For my Calc III homework problem, I am unsure how to determine the region of integration. The problem is fairly simple. $$ \text{Find the volume of the solid bounded by the planes }x = 0\text{, }y = ...
1
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0answers
22 views

Converting cartesian to polar integral

I feel like I almost have a grasp on regions of integration, I am a bit frustrated that I haven't fully gotten it but because I feel like I'm almost there. In this particular homework problem I have a ...
-3
votes
2answers
35 views

How to find the area under a semicircle using integration? [closed]

How would I go about finding the area under a semicircle? I know that to use integration the formula is $\int_a^b f(x) \mathrm{d}x.$However, when I put this into my graphing calculator it doesn't ...
1
vote
1answer
55 views

Finding the general integrals of functions like $\frac1{x^n+1}$, $\cos^nx$. [closed]

This question is just a soft question, about can we compute a general formula for everything? Or it has some restrictions? Like $\int x^ndx=\frac{x^{n+1}}{n+1}+C$. I am not able to deduce a formula ...
1
vote
1answer
68 views

Stuck on integration question

The curve in the picture shown has equation $y=bx(x-2)$ (a) Find b given that the shaded area is 4 units$^2$ (b) Find the x-coordinate of the point A if the line OA divides the shaded area into ...
-2
votes
2answers
39 views

How to write the integral $\int_R 5(x+y)\ dy\ dx$ where $R$ is the region bounded by $y=\frac{1}{7}x$, $x=6$ and the $x$-axis?

I have the integral $$\int_R 5(x+y)\ dy\ dx$$ where the region $R$ is bounded by $y=\frac{1}{7}x$, $x=6$ and the $x$-axis. I don't know how to write this problem exactly. Could anyone edit my ...
0
votes
2answers
16 views

Setup region of integration for polar coordinates

I've been working on a homework set for Calc III, right now we're emphasizing double integration and polar integrals. I keep having problems conceptualizing where to actually create my region of ...
1
vote
2answers
32 views

Find total area under infinite curves

My question is finding the total area covered by curves, such as the total area every curve in the following picture covers (from 100 on y axis to 200 on x axis): In my case, the curves are ...
0
votes
0answers
28 views

Volume formed by Rotating about an Equation of Line

Find the volume formed by revolving the triangle whose vertices are $\{(1,1),(2,4),(3,1)\}$ about the line $2x - 5y = 10$. The Answer is $56$. To solve the area of the triangle, I use Heron's ...
3
votes
1answer
96 views

How to prove $\int_{0}^{\infty}\frac{e^{-\left(\sqrt{x}-a/\sqrt{x}\right)^2}}{\sqrt{x}}dx=\sqrt{\pi},\,a>0$?

We know that $$\Gamma\left(\frac{1}{2}\right)=\int_{0}^{\infty}\frac{e^{-x}}{\sqrt{x}}dx=\sqrt{\pi} $$ but it seems that, for every $a>0 $ we have ...
1
vote
1answer
98 views

Calculating in closed form $\int_0^1 \log(x)\left(\frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}}\right)^2 \,dx$

What real tools excepting the ones provided here Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $ would you like to recommend? I'm not against them, they might ...
0
votes
0answers
25 views

Can you prove that the integral below, with a vectorial field, is zero?

If $\vec{J}(\vec{r})$ is a vector field limited in infinity. Prove that the integral below is zero: \begin{equation} ...
2
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0answers
56 views

How to integrate the following sum?

I'm currently trying to show: $$ \int_0^1{\int_0^y{\sum_{n=0}^{\infty}\left(\frac{1}{10^{n+1}x(1-x)}\left(9+\frac{1}{1-x^{10^n}}-\frac{10}{1-x^{10^{n+1}}}\right)\right)dx}dy}=\frac{10}{99}\log(10) $$ ...
1
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2answers
44 views

How to integrate a function of form f (x)/g (x) $\frac {(ax^n+b) dx}{cx^m+d}$

How to find $\int \frac {(ax^n+b) dx}{cx^m+d}$ (where m>n and m,n are natural numbers,and a,b,c,d are integers) ?
10
votes
2answers
145 views

Integral $\int_0^1\frac{\log(x)\log(1+x)}{\sqrt{1-x}}\,dx$

I'm trying to evaluate this definite integral: $$\int_0^1\frac{\log(x) \log(1+x)}{\sqrt{1-x}} dx$$ It's clear that the result can be expressed in terms of derivatives of a hypergeometric function with ...
0
votes
0answers
18 views

How to perform the following integration using dblquad in MATLAB

I am trying to perform the following integration in MATLAB \begin{equation} \begin{split} F &= @(x,y)(e^{(-0.5([x - \mu_1 \hspace{5pt}y-\mu_2])\Sigma^{-1}([x - \mu_1 ...
0
votes
3answers
21 views

Double integrals volume

Find the volume below $z = 5+3y$ above the region $−5 \leqslant x \leqslant 5$, $0 \leqslant y \leqslant 25−x^2$. How do I solve this? I don't know how to make equation to solve this problem. Anyone ...
2
votes
1answer
45 views

Lim sup/inf of average value

Consider $$f(t)= \frac{1}{t} \int_{0}^t \sin(e^s) ds.$$ What is $$\mathrm{lim \ inf}_{t \rightarrow \infty} f(t)$$ and $$\mathrm{lim \ sup}_{t \rightarrow \infty} f(t)?$$ Using $u$-substitution, ...
-2
votes
0answers
29 views

Using integral to find a volume of bounded region [closed]

Math Question The base of each soild below is the region in the xy-plane bounded by the $x$-axis, the graph of $y = x^{1/2}$ and the line $x =3$. Find the volume of each solid A. Each cross-section ...
1
vote
4answers
136 views

$U_n= \int_{0}^{1}\frac{1}{1+x^{n}}dx$

$U_n= \int_{0}^{1}\frac{1}{1+x^{n}}dx$ where Find $\lim_{n\to \infty} U_n$ can i enter the limit inside ? $W_n= \int_{0}^{1}\frac{x^n}{1+x^{n}}dx$ and i established this relation by parts: $W_n= ...
1
vote
1answer
49 views

How to solve this problem using spherical coordinates system?

The question is very simple: Volume inside the solid limited by:$ (X^2+Y^2+Z^2=16), (X^2+Y^2=4)$ using SPHERICAL coordinates system. The final answer however can be checked making a "cylindrical ...
11
votes
3answers
295 views

Evaluating $\int_0^1 \frac{1}{\sqrt{\Gamma(x)}} dx$

What is the value of the following integral? $$\int_0^1 \frac{1}{\sqrt{\Gamma(x)}} \,dx$$ Here $\Gamma(x)$ is Euler's gamma function. EDIT: Can we improve the upper bound strictly smaller than $1$? ...
0
votes
0answers
33 views

Can I compute marginal distribution this way?

I have posted the same question in the Internet another website. But I did not get the answer replies. I only can come here to have a try. The math statement I put here may not be correct. You can ...
0
votes
2answers
44 views

Calculating integral using Stokes theorem and directly

Here is my task: Calculate directly and using Stokes theorem $\int_C y^2 dx+x \, dy+z \, dz$, if $C$ is intersection line of surfaces $x^2+y^2=x+y$ and $2(x^2+y^2)=z$, orientated in positive ...
2
votes
0answers
36 views

Help Integrating $I=\int\Phi\left(\frac{p}{\sqrt{q+rx}}\right)dx$

I am trying to integrate the following function involving the Normal CDF ($\Phi$). I actually need the definite integral $$\int^b_a\Phi\left(\frac{p}{\sqrt{q+rx}}\right)dx$$ for $q+ra,q+rb >0$ but ...
0
votes
4answers
23 views

Double Integral Set Up

The question was stated as follows, Evaluate the following double integral; $$ \iint_R x^3y dA $$ where R is interior of triangle with vertices (0,0), (1,0), & (1,1) . I thought for these ...
2
votes
1answer
51 views

integrals of exponential functions over the real axis

How to evaluate the integral $$ \int_{-\infty}^\infty \exp(-\sqrt{1+x^2})dx? $$ I intend to change the variable $x$ to $\tan t$ but failed... How to solve it?
1
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2answers
525 views

Explain why graph of f lies below the $x$-axis in interval $[4\pi/9,5\pi/9]$

$f(x)=(x+1)\sin(3x)$ Explain why the graph of f lies below the $x$-axis for values of $x$ in the interval $[4\pi/9, 5\pi/9]$ From what i know/understand I'd have to look at the function in two ...
2
votes
1answer
82 views

What is the general solution for integrals of the form $\int\frac{\;\ln^{m}(x+n) }{(x+n)^{b}e^{\alpha (x+n)}} dx$?

I have this integral $$\int\frac{\;\ln^{m}(x+n) }{(x+n)^{a}e^{\alpha (x+n)}} dx;\;\;m,n\in\mathbb{N_{>0}};\;\;a\in\mathbb{Q};\;\;\alpha\in\mathbb{R_{>0}}$$ I've tried to solve it with ...
0
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0answers
50 views

Contour integration from zero to infinity

When solving an improper integration from $0$ to $\infty$ which involves an even function, the integration limits can be extended from $-\infty$ to $\infty$. For example consider even function $f(x)$; ...
4
votes
5answers
73 views

Properties of $L^2(-1,1)$ functions

I want to show that there is no function $v \in L^2(-1,1)$ with $\int_{-1}^{1} v(x)\phi(x) dx = 2\phi(0)$ for all $\phi \in C^\infty_0(-1, 1)$ ($\phi$ is $0$ everywhere but $[-1,1] $). I know about ...
1
vote
0answers
18 views

Residue Theorem on an integral contains a Hankel function and a cosine function

I am trying to solve below integration; $$\int_{0}^{\infty} H_{0}^{1}(pR)\sin(pR)\frac{p}{k^2-p^2} dp$$ here $k,R$ are constants. This is related to the question link. Below shows my approach to get ...
5
votes
8answers
675 views

Evaluating $\int_{0}^{1} \sqrt{1+x^2} \text{ dx}$

I'm learning integral. Here is my homework: $$\int_0^1 \sqrt{1+x^2}\;dx$$ I think this problem solve by change $x$ to other variable. Can you tell me how please. (just direction how to solve) ...
0
votes
0answers
26 views

Help with a Lebesgue integration problem

Let $(X,m,\mu)$ be a Lebesgue measure space and $f:(X,m)\to[0,\infty]$ be a Lebesgue measurable function such that $\int_X f d\mu=3$. For each $n\in N$ consider the function ...
4
votes
1answer
40 views

Taking out absolute value on the solution to integral equation

I have this equation:$$y=2+\int_2^x (t-ty(t))dt$$ After solving it I got the answer $-\ln|1-y|=\frac {x^2} 2-2$ although the book has the same answer without the absolute value in the logarithm, why ...
1
vote
0answers
16 views

Double integral with triangular region of integration

I'm attempting to do a Calc III homework problem, and I feel like I'm on the right track, but somewhere I either mess up or set up the problem incorrectly and I don't know how or why. Here is the ...
0
votes
3answers
46 views

How to evaluate $\lim\limits_{n\to\infty} {\sin({b\over n})+\sin({2b\over n}) + \ldots + \sin({nb\over n})\over n}$ by relating it to a Riemann sum? [closed]

Evaluate the following limit by relating it to a Riemann sum: $$\lim_{n\to\infty} \frac{\sin\left(\frac{b}{n}\right)+\sin\left(\frac{2b}{n}\right) + \ldots + \sin\left(\frac{nb}{n}\right)}{n}$$
5
votes
3answers
181 views

Given the differential equation, how to solve the y function with x as the independent variable?

$y\frac{dy}{dx} = x(y^4 + 2y^2 + 1)$ $y = 1$ when $x = 4$ I tired to integrate by substitution, but it doesn't seem to work out.
1
vote
0answers
28 views

Inequality of sup and inf of an exponential function

Let $a<b<c$ and $\alpha,\beta\geq0$, and let $p:[a,c]\to(1,\infty)$ be a function. I came up with the following inequality: \begin{align*} (c-a)\inf_{a\leq t\leq ...
1
vote
3answers
47 views

Find the $\int \frac{(1-y^2)}{(1+y^2)}dy$

$\int \frac{(1-y^2)}{(1+y^2)}dy$ first I tried to divide then I got 1-$\frac{2y^2}{1+y^2}$ and i still can't integrate it.
1
vote
3answers
48 views

Solving the differential equation $dr=(r\cos\theta +r\sin\theta)d\theta$

$dr=(r\cos\theta +r\sin\theta)d\theta$ In my book this is under separation of variables then i tried to factor out r and divide both sides then integrate both sides but where can i find my $C_1$? I ...
2
votes
2answers
154 views

Monte Carlo Importance Sampling

I am reading the book on Monte Carlo by Sobol (A Primer for the Monte Carlo Method). In the section on Importance Sampling, he writes: $I = \int_a^b g(x) \: dx$ "to compute this integral, we could ...