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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2
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0answers
38 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 ...
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
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 $f(x)$; ...
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 ...
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= ...
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
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 ...
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 = ...
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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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 ...
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 ...
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: $$\lim_{n\to\infty} \frac{\sin\left(\frac{b}{n}\right)+\sin\left(\frac{2b}{n}\right) + \ldots + \sin\left(\frac{nb}{n}\right)}{n}$$
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 ...
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
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 ...
0
votes
0answers
111 views

A double integral consisted of hypergeometric functions [closed]

Calculate in closed form $$\small\int _0^1\int _0^{\infty }\left(-\frac{9 \sqrt{\frac{3}{\pi }} \Gamma \left(\frac{4}{3}\right) \Gamma \left(\frac{5}{3}\right) \, ...
0
votes
0answers
21 views

What is the volume of this set?

Given $b \gt 0$ and $M:=\{ (x,y,z) \in \mathbb R^3 : 0 \lt x \lt b \land y^2+z^2 \lt e^{-2x}\}$ What is the volume of $M$ ? My main problem is to find out the limits of the integrals. $0 \lt x \lt ...
-1
votes
0answers
43 views

Surface integral - area and volume [closed]

Here is my task: Ball S with radius R and center in coordinate beginning is given. Let $\sum $ be its boundary. a) Calculate area of sphere $\sum $ and volume of ball S, which are located between ...
0
votes
0answers
46 views

A divergent sequence of integrals

Let {$s_{n}$} be a sequence of increasing step functions that converges pointwise to a limit function $f$ on an interval $I$ that is unbounded. $f(x)\ge 1$ almost everywhere on $I$ . ...
3
votes
3answers
103 views

Is this inequality true? If yes, for what functions?

Let $B=B(0,1)\subset \mathbb R^2$. Let $u$ be a radially symmetric differentiable function on $B$ and $v=Ax+b$ be a linear function where $A$ is a $2\times 2$ matrix satisfies $A=-A^T$, and ...
0
votes
5answers
84 views

What is the mistake in doing integration by this method?

Integration Of a given function can be found out in many ways, For a specific function ∫1/xlogx, if we do integration by parts (∫f(x) g(x)= f(x) ∫ g(x)- ∫ [d/dx (f(x)) ∫g(x)] dx ) we get this way ...
8
votes
3answers
110 views

Logarithmic Integral II

While reviewing an old calculus book the following integral was assigned: \begin{align} \int_{0}^{1} \left( x^{a-1} - x^{n-a-1} \right) \, \frac{\ln^{2}x \, dx}{1-x^{n}} = \frac{2 \, \pi^{3} \, ...
1
vote
2answers
24 views

what is the value of $\theta$ used in calculate volume bounded by $z=x^2+y^2$ and $x^2+y^2=2x$

This is an example from my textbook, it explain everything well except the reason why $\frac{\pi}{2} < \theta < \frac{-\pi}{2}$ but not $2\pi < \theta < 0$. It's not explained and I can't ...
1
vote
1answer
36 views

Fourier series of constant on $2\pi$ intervals

I want to find a fourier expansion of only sines representing $g(x) = 1$ on the interval $[0, \pi]$. So I extend the function on $[-\pi, \pi]$ such that it is odd, and calculate $$b_k = \frac 1\pi ...
1
vote
1answer
44 views

Line integrals - Surface area

Here is my task: Calculate surface area of $2(x^{2}+y^{2})^{2}=xy$ between surface $x^{2}+y^{2}=z$ and $z=0$. Here is my attempt to solve this problem. Firstly, I transformed line ...
2
votes
5answers
66 views

$U_n=\int_{n^2+n+1}^{n^2+1}\frac{\tan^{-1}x}{(x)^{0.5}}dx$ .

$U_n= \int_{n^2+n+1}^{n^2+1}\frac{\tan^{-1}x}{(x)^{0.5}}dx$ where Find $\lim_{n\to \infty} U_n$ without finding the integration I don't know how to start
-1
votes
1answer
66 views

Find the x-coordinates of two other points of inflection of $f(x)= \int \frac{x+1}{x^2+1}$, given there is an inflection point at $(1,1) $

$$f(x)= \int {\frac{x+1}{x^2+1}}$$ I have to find the x-coordinates of two other points of inflection, given there is an inflection point at (1,1). My approach is to differentiate the equation, and ...
2
votes
1answer
23 views

Relating Integration by Substitution to Change of Variables Theorem

I'm having trouble relating the change of variables theorem from measure theory to the integration by substitution formula taught in Calculus. I've always thought they were basically saying the same ...
3
votes
1answer
43 views

Solving an Iterated Integral

Given the iterated integral: $$\int^{\sqrt{2}}_{-\sqrt{2}}\int^{\sqrt{2-x^2}}_{-\sqrt{2-x^2}}\int^{\sqrt{4-x^2-y^2}}_{\sqrt{x^2+y^2}}{\left(x^2+y^2+z^2\right)^{3/2}}dzdydx$$ Now, my question is, ...
1
vote
1answer
82 views

Calculating in closed form an integral in Airy function

Can we hope for a nice closed form for the integral below? $$\int_0^1 \frac{\displaystyle \text{Ai}\left(-\frac{t}{2^{2/3} \sqrt[3]{3-3 t}}\right)^2+\text{Bi}\left(-\frac{t}{2^{2/3} \sqrt[3]{3-3 ...
0
votes
0answers
21 views

How to evaluate integral within an integral with symbolic limit in Matlab

I want to evaluate following integral which contains another integral inside. spol, which is a function of theta,is the lower limit of internal integral. The internal integral then will be multiplied ...
0
votes
3answers
35 views

Fundamental Theorem of Calculus and the left endpoint of the interval.

Going through Stephen Abbot's Understanding Analysis he gave the following as the second statement of the Fundamental Theorem of Calculus: Let $g:[a,b]\to\mathbb{R}$ be integrable and define ...
2
votes
1answer
80 views

In integral, would it make any sense to write dx as denominator?

Does it make any sense to write something like this? $$ \int \frac 1{dx} $$ I think I saw it somewhere, but since I can not find anything similar on wikipedia and wolfram, I have to assume that this ...
1
vote
0answers
47 views

integral from gradshteyn and ryzhik

I'm interested in evaluating the integral $$ \int_{a}^\infty e^{-x\cosh\alpha}\,K_{\nu}(x\sinh\alpha)\,\frac{dx}{x}, $$ where $a>0$ and $\nu$ is purely imaginary. Here $K$ denotes the MacDonald ...
0
votes
3answers
43 views

How do I solve this deceleration problem?

Question: A car is traveling at 100km/hr, when the driver sees an accident 80 meters ahead. What constant deceleration is required to stop the car in time to avoid a pileup? So far I have approached ...
2
votes
1answer
34 views

Integral of square of variable in expression in square root in denominator

I'm having trouble with following integral $$ \int \frac{1}{\sqrt{1+4x^2+4y^2}} dy $$ that according to WolframAlpha is $$ \frac{1}{2} \ln( \sqrt{1+4x^2+4y^2} + 2y) + c $$ which I can verify but I ...
0
votes
2answers
24 views

Moment of inertia about the origin of an ellipsoid?

Find the moment of inertia about the origin of an ellipsoid. Heres what I did but I believe it is incorrect: $$I_o= \iiint_{V_e}{(x^2 +y^2 +z^2)\rho dx dy dz} $$ Making Substitution of $aX=x \ bY=y \ ...
6
votes
0answers
81 views

Can you add new functions to the set of elementary functions such that every function has an anti-derivative?

Its fairly well known that not every elementary function has an elementary anti-derivative. The common examples of this are $\exp(-x^2)$ and $\sin(x)/x$. The general workaround to this problem is to ...
0
votes
2answers
49 views

show that function is odd

Consider a function $f \in C([-1,1],\mathbb{R})$ and suppose that $$\int_{-1}^{1} f(t)t^{2n}dt=0$$ for all $n \in \mathbb{N}_0$. I want to show that under this assumption the function $f$ has to be ...
1
vote
2answers
43 views

How do I solve a double integral with an absolute value?

Given the following integral $$\int_{y=0}^1 \int_{x=0}^1|x-y|(6x^2y) \, dx \, dy$$ how do I change the limits of integration? According to my textbook, it is $$\int_{y=0}^1 ...
3
votes
3answers
82 views

How to integrate $\int \frac{\arctan x}{x^4} dx$?

I have written the integral as $\int x^{-4} \arctan x dx$. Then, by applying by parts, I got $-3\dfrac{\arctan x}{x^3} + 3\int \dfrac{1}{x^3(1 + x^2)} dx$. Now, how can I solve the later integral? Is ...
2
votes
2answers
97 views

Integration of $\int \frac{(1 + x)\sin x}{(x^2 +2 x)\cos^2 x-(1 + x)\sin2x}dx$

The integral is $$\int \dfrac{(1 + x)\sin x}{(x^2 + 2x)\cos^2 x-(1 + x)\sin2x}dx.$$I've tried the problem by first multiplying both the numerator and denominator by $\sec^2 x$ but couldn't do justice. ...
0
votes
4answers
91 views

Primitive of $\left(x +\sqrt{1 + x^2}\right)^n$

How to find a primitive of $\left(x +\sqrt{1 + x^2}\right)^n$? I have started it by parts but it never could end in a good position.
0
votes
0answers
27 views

Geometric interperation of line integral - example

I have hard times figuring out geometric interpretation of line integrals. Here is one example from my book: Calculate area of cylinder $x^{2}+y^{2}=ax$ sliced with sphere $x^{2}+y^{2}+z^{2}=a^{2}$. ...
5
votes
3answers
73 views

If $\lim_{n\to\infty}\frac{1^a+2^a+…+n^a}{(n+1)^{a-1}.((na+1)+(na+2)+…+(na+n))}=\frac{1}{60}$, Find the value of a

If $$\lim_{n\to\infty}\frac{1^a+2^a+...+n^a}{(n+1)^{a-1}\cdot((na+1)+(na+2)+...+(na+n))}=\frac{1}{60}$$ Find the value of a. Attempt: I solved it using two methods each giving me different answers. ...
1
vote
4answers
131 views

Explain the integral of $1/x = \ln |x| + \mathrm{C}$ graphically as sum of area?

I am unable to interpret the integral $$\int {1\over x}{\rm d}x=\ln|x|+\mathrm{C}$$ Graphically as area under the curve of $1/x$ (as the definition of the integral). Can somebody please ...
1
vote
1answer
35 views

Sequence of integrals of positive function

Let $f(x)$ be a function positive almost everywhere on $X$. Let $A_n$ be a sequence of subsets of $X$ such that $m(A_n) > c> 0$ for all $n$, where $c$ is some constant, and $m$ denotes the ...
2
votes
1answer
42 views

Residue theorem on even function integration

I need to integrate below function; $$\int_{-\infty}^{\infty} \frac{\sin(pR)}{R}\frac{p}{k^2-p^2} dp$$ here $k,R$ are constants. Since this is an even function of $p$, I tried applying the residue ...