All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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-1
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2answers
38 views

Proving the Volume of an Ellipsoid

This is the question: The solid generated by rotating the region inside the ellipse with equation $$ \left( \frac{x}{a} \right)^2 + \left( \frac{y}{b} \right)^2 = 1 $$ around the $x$-axis is ...
0
votes
2answers
41 views

How to proof $ \underset{n\to\infty}\lim \int_0^1 f(t)\, \mathrm{sgn}\big(\sin (2\pi n t)\big)\,dt = 0$?

I was wondering if you could provide hints which could lead me to a rigorous proof for the following: Given $\,f\in L^1([0,1])$, then $$ \underset{n\to\infty}\lim \int_0^1 f(t)\, ...
1
vote
1answer
18 views

Mean value theorem for integrals: how does the sign matter?

The mean value theorem is that $\int_a^b f(x)g(x) dx = f(\xi) \int_a^b g(x) dx$ for some $\xi \in [a,b]$ if g(x) does not change sign on $[a,b]$. However, I can't see why the sign matters. There is a ...
0
votes
1answer
26 views

Integrals in Reverse

I'm asked what solid this integral represents(the integral is used to obtain the volume of a solid, we are given this). I see that since we have the 2pi, this is probably a volume obtained by using ...
2
votes
2answers
45 views

Absolute continuity under the integral

Let $f:[0,T]\times \Omega \to \mathbb{R}$ where $\Omega$ is some bounded compact space. Let $t \mapsto f(t,x)$ be absolutely continuous. Is then $$t \mapsto \int_\Omega f(t,x)\;dx$$ also absolutely ...
0
votes
1answer
17 views

Integration in d-dimensional spherical coordinates

Can someone please tell me why $$\left(\int_0^\infty dr\, e^{-r^2}\right)^d=\int_0^\infty dr\,r^{d-1}S_d e^{-r^2}?$$ Why doesn't $d$ end up joining the exponential?
1
vote
1answer
64 views

Cauchy Schwarz in an integral with distributions.

I am working with energy methods for PDE's and I have a expression of the following form: \begin{equation} \int f \phi\phi_{j} \end{equation} under the conditions that $f\in L^{\infty}, \phi\in ...
1
vote
0answers
39 views

Closed-form solution for the integral of the ratio of two sums?

Is there a closed-form solution for this integral? $$\int_0^t\frac{\sum_{k=1}^Nr_{1k}\exp(ixr_{2k})}{\sum_{k=1}^N\exp(ixr_{2k})}dx$$ for given $r_{11},...,r_{1N},r_{21},...,r_{2N} \in \mathbb{R}$
-3
votes
1answer
34 views

Tripe integral, with the integration of $\sin(x+y^2)$ [closed]

how to calculate the integral below: $$I=\iiint_{\Omega} |\sin(x+y^2)|dxdydz$$ where $\Omega$ is the zone whose edge is $x=0$,$y=0$,$x=\pi$,$y=\pi/2$ and $z=y$. Thanks very much.
1
vote
1answer
26 views

A list of techniques to try when confronted with an integral?

I didn't solve any integrals for 2 months, and now I have lost most of my reflexes. What is a good checklist of techniques to try, and in which particular order ? The purpose is to explore every ...
1
vote
1answer
31 views

Prove that if expectations agree on a Pi-System, then they agree on the Sigma-Algebra generated by the Pi-System.

If $\mathscr{G}$ is a sub-$\sigma$-algebra of $\mathscr{F}$ and if X $\in L^1 (\Omega, \mathscr{F}, P)$ and if Y $\in L^1 (\Omega, \mathscr{G}, P)$ and $E(X;G) = E(Y;G)^{*} \forall G \in \mathscr{I}$ ...
0
votes
0answers
19 views

Cauchy Principal Value Example

I wish to evaluate the following integral using the Cauchy Principal Value $$\int_{\omega_1}^{\omega_2} \dfrac{\omega'}{\omega'^2 - \omega^2}d{\omega'}$$ given that $$\int ...
2
votes
2answers
25 views

Multiple answer for integration of a function?

Q. $\int \left(\frac{sin2x}{sin^4x+cos^4x}\right)\:dx$ My method: $$\int \:\left(\frac{sin2x}{sin^4x+cos^4x}\right)\:dx=\int ...
0
votes
1answer
61 views

How can I find $\int \dfrac{ dx}{1+x^4}$ [duplicate]

I saw the solution on a website and I didn't understand most of it. I would appreciate if someone could please explain it using baby steps for me. Thanks,
2
votes
1answer
50 views

Using the Fourier Series of $f(t)=(t-\frac{1}{2})^{2}$ to deduce the sum $\sum_{n=1}^{\infty }\frac{1}{n^{2}}$?

So this is a question in one of the previous tests: My approach (if you want just skip to step 3.):$$$$ 1. Formulation of the problem and calculating the constant term of the series $a_o$ I ...
0
votes
0answers
23 views

Finding the integral of arccos(x) to be 0.5 [closed]

The integral from k to 0 of arccos(x) with respect to x is 0.5. Ok, so I came as far as: karccos(k) - sqrt(1-k^2) = -0.5 But from here, if the right path, I failed to find the solution. Any help ...
0
votes
1answer
76 views

About Riemann integrability

I need to prove if $f$ is continuous on an interval $I$, then its Riemann integral exists. It is hard for me because it is an interval and not closed interval. Can anyone give me some answers or ...
1
vote
1answer
23 views

solving this integral

$$\int_0^1 \int_0^\eta \phi \, d\eta' d\eta$$ $$\phi=\eta-3\eta^3+2\eta^4$$ This is from a fluid mechnanics paper. Can someone solve it and tell me what dn' means since n is a variable already? ...
0
votes
1answer
42 views

Use complex number to solve this equation $\int e ^{3x} cos x dx$?

I can solve it another way, but am not sure how to use complex numbers to solve it. Thanks for your help
-2
votes
2answers
22 views

Can someone ple help me answer ths questions. Calculus || [closed]

Consider curves: f(x) = x^3 and g(x) = x^2 (a) Find all points of intersection of these two curves. (b) Find the area between these two curves.
0
votes
1answer
24 views

Finding Volume with Solids of Revolution [closed]

Find the volume of the solid obtained by revolving $y=8-x^3$ about the $x$-axis from $x = 0$ to $x = 2.$
0
votes
0answers
18 views

Checking some work on an expectation value problem

I am working on a pretty simple problem (or so it seems it should be) from Griffith's QM text. The problem states: for the probability density function $\rho (x) = Ae^{-\lambda(x-a)^2}$ a) find A ...
-1
votes
2answers
45 views

Simple integration question y prime and y^2 [closed]

How does one integrate $y'/y^2$ in terms of y? I'm stumped.
0
votes
1answer
17 views

Finding volume of solid using two methods

The question is: The region bounded by $y=\frac{1}{x}, y=0, x=1, x=2$ is rotated about the $y$-axis, thus creating a solid. Compute the volume using the Shell and Slicing method. This is what I have ...
0
votes
0answers
46 views

Integral Problem

A particle of mass $m$ is projected towards a point $O$ with initial speed $\dfrac{\sqrt{5}}{3}$ m/s from a point $P$, where $\overline{OP}$ is 3 meters. The particle is repelled from $O$ by a force ...
0
votes
2answers
50 views

Is there an easier way to integrate $\int \frac{d\, x}{\sin^2{x}}$?

So, hey. I've made it like this: $ \tan{\frac{x}{2}} = t, x=2\arctan{t}, d\, x =\frac{2t\,d\,t}{1+t^2}$, therefore $\int\frac{d\, x}{\sin^2{x}} = \int\frac{1+t^2}{2t^2}d\,t$, which gives us ...
0
votes
1answer
56 views

Prove: If $f,g$ agree on $A\subseteq [a,b]$ then $\int_a^b f = \int_a^b g$.

Prove: If $f,g$ agree on $A\subseteq [a,b]$ then $\int_a^b f = \int_a^b g$. In short, the proof starts with: We'll choose partitions, $\Pi_n$ such that $\Pi_n \to \infty$. And they key in this ...
0
votes
2answers
47 views

Integration by substitution: What formula can I refer to?

When I am trying to integrate a composite function $f(g(x))$ that is multiplied by $g'(x)$, then there's a formula for that in my book. It's simply $F(g(x)) + C$. But what if $g'(x)$ isn't there? ...
1
vote
1answer
32 views

Different answer when simplifying before integrating

I have been trying to get my head around this for some time now... I solve the same integral in two ways but get two different solutions. Since there can't (surely) be any sort of ambiguity when ...
3
votes
1answer
53 views

Stuck on tough integral

I am trying to solve what looked like a simple integral but I got a bit stuck. The integral is : \begin{equation} \int_0^x \frac{ab(1-e^{-ct})}{d-\frac{b(1-e^{-ct})}{c}}dt \end{equation} I tried ...
0
votes
0answers
23 views

Exchanging the limit of an integral with a finite sum

So, in general, I can get this value: $$\lim_{a \to \text{a constant}}{ \int{ \left( \sum_{x=x_1}^{x_2}{ f(a,x) }da \right)} } \tag{1}$$ What I'm after is this: $$\sum_{x=x_1}^{x_2}{\left( \lim_{a ...
0
votes
0answers
22 views

Comparing values of integral after and before applying a diffeomorphim

Let $\phi:\mathbb{S}^1\to\mathbb{S}^1$ be a diffeomorphim such that its Jacobian satisfies $\int_{\mathbb{S}^1}\operatorname{Jacobian}(\phi)=2\pi$, and let $f:\mathbb{S}^1\to\mathbb{S}^1$ be a smooth ...
0
votes
3answers
93 views

An estimate for the lower Riemann sum for the derivative of a differentiable function

I have been looking at this question for the past couple days and keep walking away from it as I have no clue where to begin. Would anyone be able to point me in the right direction or give me any ...
1
vote
1answer
28 views

Lesbegue integration w.r.t scaled measure

I was wondering if we have a measure $\mu$ and $a \in \mathbb{R}$ with $\lambda = a\mu$ if we get : \begin{align} \int fd\lambda = a\int fd\mu \end{align} It seems to work for the basic definition of ...
16
votes
3answers
185 views

Finding integral of a function

I have stumbled upon an exercise that reads thus: $$\int\limits_{-\infty}^0\frac{x^x}{x^3-1}\mathrm{d}x=\frac{2\sqrt3}{9}\pi,$$ and I am guessing it is asking to prove the above equality. ...
2
votes
2answers
71 views

how to integrate $\sqrt{1-x^{2/3}}$

i'm facing the following problem: find the area of the set $M=\{(x,y): |x|^{\frac{2}{3}}+|y|^{\frac{2}{3}}\le 1\}$ using integration i thought about only integrate where x,y both $\ge0$ and multiply ...
8
votes
1answer
129 views

Find the closed form of $\sum_{n=1}^{\infty} \frac{H_{ n}}{2^nn^4}$

One of the possible ways of computing the series is to obtain the generating function, but this might be a tedious, hard work, pretty hard to obtain. What would you propose then? ...
0
votes
2answers
42 views

Prove that $\displaystyle\int_0^\pi \frac{dx}{a^2 \cos^2 x+b^2 \sin^2 x} = \frac{\pi}{ab}$ [closed]

I need solution for Prove that $\displaystyle\int_0^\pi \frac{dx}{a^2 \cos^2 x+b^2 \sin^2 x} = \frac{\pi}{ab}$ Help me to find it as early as possible
6
votes
2answers
71 views

A pseudometric on the space of the measurable functions is complete

I'm working in the following exercise: Suppose $(X, \mathcal A, \mu)$ is a finite measure space and suppose $\mathcal F$ is the set of all $\mathcal A$-measurable functions $f: X \rightarrow \mathbb ...
0
votes
1answer
25 views

How I integrate this function? with Delta Fuction

here is $$∫_0^l F(x,ζ)ϕ(x)dxdζ$$ with this $$F(x,ζ)=Q(ζ)δ(x-x0) e^{-rx}$$ Thanks for your help!
5
votes
0answers
90 views
+50

Closed form of integrals containing double exponentials

Are there closed forms for the following integrals? $$\begin{align} I_1(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y)}{y^2+\pi^2} dy, \\ I_2(w) & = \int_{-\infty}^{\infty} ...
1
vote
1answer
51 views

Autonomous differential equation

Let $f: \Bbb R \to \Bbb R$ and $x_0 \in \Bbb R$, such that $f(x_0)> 0 $, and assume that $x(t)$ is the solution of $x'=f(x)$, such that $x(0)=x_0$. If $f(x) > 0$ then $x(t)$ is defined for all ...
3
votes
0answers
75 views
+150

How to take this Grassmann integral?

I'm trying to reconstruct and understand what is explained in a paragraph of this paper. I am trying to check if the method they describe actually gives us the Laughlin state. The integral I'm facing ...
1
vote
0answers
31 views

Evaluating convoluted integrals of complex exponentional and rational

I want to evaluate the following integral: \begin{equation} f_{abcd}(t) = \int_{-\infty}^{\infty}d\lambda\int_0^{t-\lambda} d\tau \frac{e^{i a \tau}}{ (b+i \tau)^{5/2} } \int_0^{t-\lambda} d\tau ...
0
votes
3answers
51 views

How to find the integral of $(1-|\tau|)\cos(\omega\tau)e^{-j\omega\tau}$

I have a function that need calculate the integral. Could you help me to find it. Thank you so much $$f(\omega)=\int_{-1}^1(1-|\tau|)\cos(\omega\tau)e^{-j\omega\tau}d\tau$$ where $\omega$ is constant. ...
2
votes
1answer
48 views

Switching $\int$ and $\sum$ proof

Been reading through this proof which seems incorrect: Let $f_n$ be continuous on the curve $C$ and $\sum f_n$ converge uniformly on $C$. Then $\sum\int_Cf_n(z)dz=\int_C\sum f_n(z)dz$ PROOF: ...
0
votes
2answers
84 views

How to write this integral in a nice way?

I have a function $f(a,b):= \int_{-1}^{1} e^{i (ax+bx^2)}dx$ with $(a,b) \in \mathbb{R}^2 \backslash \{(0,0)\}$ and now I want to find out what $|f(a,b)|^2$ is. Is there a way to write this in a ...
2
votes
1answer
40 views

$\int_0^{2 \pi} \cos(x)e^{i (a \cos(x) + b \cos^2(x)} dx$ and $\int_0^{2 \pi} \cos^2(x)e^{i (a \cos(x) + b \cos^2(x)} dx$

I am currently dealing with the two integrals in the title and I want to find out, when their real part of their imaginary part vanishes ( so for which constellation of $(a,b) \in \mathbb{R}^2 ...
7
votes
1answer
112 views

Compute polylog of order 3 at $\frac{1}{2}$

How to compute the following series: $$\sum_{n=1}^{\infty}\frac{1}{2^nn^3}$$ I am aware this equals polylog of order 3 at $\frac{1}{2}$, but how to prove it using integral or Euler sum only (without ...
0
votes
0answers
63 views

Is there a 4 pointed star that is regular?

I am studying about the area of a 4 pointed star, I wonder if there is really a 4 pointed star that is regular? what could be the characteristics of a regular 4 pointed star? that is how i ...