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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0answers
36 views

Proving that an infinite series equal a finite series

Suppose we have a function $f(z)$, which has $m$ isolated singularities, which are non-integers (say, $z_1$, $z_2$,...,$z_m$). Define $H(z):=\frac{\pi f(z)}{\sin(\pi z)}$. Assume that there exists a ...
3
votes
1answer
25 views

Does it follow that $\{f_n\}$ is uniformly integrable?

Suppose $\mu$ is a finite measure, $f_n \to f$ almost everywhere, each $f_n$ is integrable, $f$ is integrable, and $\int |f_n - f| \to 0$. Does it follow that $\{f_n\}$ is uniformly integrable?
0
votes
2answers
34 views

Integral of the level curve $ye^{2x}$

$\int_{0}^{2}\int_{0}^{4} ye^{2x}$ $dydx$ The book says the answer is $32(e^4-1)$ I had to do u-substitution when doing it, maybe that's where I went wrong. First integrating with respect to y, ...
1
vote
1answer
64 views

How to prove that : $\int_{2}^{x}\frac{dO(te^{-c\sqrt{\log t}})}{t\log t}=O(1)$ [closed]

$\displaystyle{ \mbox{How I can prove that}\ \int_{2}^{x}{\,\mathrm{d}\alpha\left(t\right) \over t\,\log\left(t\right)} = \,\mathrm{O}\left(1\right)\ ?,}\quad $ where $\alpha\left(t\right) = \,\mathrm{...
4
votes
3answers
120 views

What is the value of $I=\lim_{n \to \infty} \int_0^1 {{1 + nx^2}\over{(1 + x^2)^n}} \log(2 + \cos(x/n))\,dx.$?

Find the integral $I$.....it looks like a good problem which I was not able to solve ....please help... $$I=\lim_{n \to \infty} \int_0^1 {{1 + nx^2}\over{(1 + x^2)^n}} \log(2 + \cos(x/n))\,dx.$$
6
votes
1answer
71 views

Do we necessarily have that $\int g\,d\mu_n \to \int_0^1 g\,dx$?

Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0, 1]$. Suppose $\mu_n$ are finite measures on $([0, 1], \mathcal{B})$ such that $\int f\,d\mu_n \to \int_0^1 f\,dx$ whenever $f$ is a real-valued ...
8
votes
2answers
50 views

$\{f_n\}$ is uniformly integrable if and only if $\sup_n \int |f_n|\,d\mu < \infty$ and $\{f_n\}$ is uniformly absolutely continuous?

Let $(X, \mathcal{A}, \mu)$ be a measure space. A family of measurable functions $\{f_n\}$ is uniformly integrable if given $\epsilon$ there exists $M$ such that$$\int_{\{x : |f_n(x)| > M\}} |f_n(x)...
4
votes
1answer
27 views

Countable collection of Borel subsets of $[0, 1]$, exists subsequence where $\int_A f_{n_j}(x)\,dx$ converges for each $i$?

Let $\{f_n\}$ be a sequence of measurable real-valued functions on $[0, 1]$ that is uniformly bounded. How do I see that if $\{A_i\}$ is a countable collection of Borel subsets of $[0, 1]$, then there ...
1
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3answers
37 views

How do I see that if $A$ is a Borel subset of $[0, 1]$, then there exists a subsequence $n_j$ such that $\int_A f_{n_j}(x)\,dx$ converges?

Let $\{f_n\}$ be a sequence of measurable real-valued functions on $[0, 1]$ that is uniformly bounded. How do I see that if $A$ is a Borel subset of $[0, 1]$, then there exists a subsequence $n_j$ ...
1
vote
1answer
17 views

Applying the Cartesian Coordinate

Find the surface area of the portion of $2x+y+z=8$ in the first octant. I know that $f(x,y)=8-y-2x$, $x=0$ to $4$ and $y= 0$ to $8-2x$, but I'm having trouble solving the problem in Cartesian form. ...
2
votes
1answer
21 views

seq. of nonneg. Lebesgue measurable functions on $\mathbb{R}$, have $\limsup_{n \to \infty} \int f_n\,dx \le \int \limsup_{n \to \infty} f_n\,dx$?

Let $f_n$ be a sequence of nonnegative Lebesgue measurable functions on $\mathbb{R}$. Is it necessarily true that$$\limsup_{n \to \infty} \int f_n\,dx \le \int \limsup_{n \to \infty} f_n\,dx?$$If not, ...
-1
votes
0answers
26 views

Finding the area under a 5th degree polynomial curve given the polynomial fitting?

Using an outdated graphing program, I have a 5th degree polynomial fitted curve for my data (protein level in a treatment). I would like to find the integral of the curve given I only have the 'p0-p5' ...
1
vote
0answers
39 views

Evaluate $\int _0 ^4 \int _0 ^{\sqrt {16-x^2}} \int _{\sqrt {x^2+y^2}} ^4 \sqrt {x^2+y^2+z^2} \ \Bbb d z \ \Bbb d y \ \Bbb d x$ [closed]

Evaluate $$\int \limits _0 ^4 \int \limits _0 ^{\sqrt {16-x^2}} \int \limits _{\sqrt {x^2+y^2}} ^4 \sqrt {x^2+y^2+z^2} \ \Bbb d z \ \Bbb d y \ \Bbb d x .$$ I've always had some difficulty using ...
0
votes
0answers
29 views

Integral of a solid, which lies inside a cylinder

Let $S$ be the surface that lies inside the cylinder $x^2+y^2=16$ and between the planes $z=0$ and $z=5$ , There is a vectorfield $\mathbf{F}$ given by: $\mathbf{F}(x,y,z)= -x^3\mathbf{i}-y^3\mathbf{j}...
5
votes
2answers
61 views

$f: \mathbb{R} \to \mathbb{R}$ integrable, $F(x) = \int_a^x f(y)\,dy$, $F$ necessarily continuous

Suppose $f: \mathbb{R} \to \mathbb{R}$ is integrable, and we define$$F(x) = \int_a^x f(y)\,dy.$$Why does it follow that $F$ is necessarily a continuous function?
0
votes
2answers
30 views

Compute $\int _Lye^{xy}dx +xe^{xy}dy$ from $(0,0)$ to $(2,1)$, $L : x = 2ye^y$

Compute $$\int _Lye^{xy}dx +xe^{xy}dy$$ from $(0,0)$ to $(2,1)$, $L : x = 2ye^y$ What I've tried: If I input $0,0$ then I get $(0,0)$, if I input $(2,1)$ I get $(2e,1)$. I also know that $P'_y = Q'...
5
votes
5answers
81 views

Integral of the product $x^n e^x$

I would be very pleased if you could give me your opinion about this way of integrating the following expression. I think that it has no issues, but just wanted to confirm: $$ \int x^n e^x dx $$ $$ e^...
0
votes
1answer
24 views

Finding a two-dimensional chain

Let $$T=\{(x,y,z,w,)\in R^4:x^2+y^2=z^2+w^2=\frac{1}{\sqrt 2}\}$$ and $$\omega=dx\land dy + dz\land dw$$ in $\mathbb R^4$. How do I find a two-dimensional chain $C$ where $T$ is its trace? And how can ...
0
votes
0answers
21 views

Using 2D Parseval-Plancheler theorem to solve an equation

In the context of a digital communications problem I have to solve the following equation with respect to $\tilde{\tau}$: \begin{eqnarray*} &&Im\Big\{\Big(\int\limits_0^{T_0}r^{*}(t)g(t-\tilde{...
0
votes
0answers
25 views

Finding area of hypocycloids (without integration)

I have been trying to find the area of hypocycloids, I understand how to do it with integration. But the thing is I wanna find some other method for finding its area. In one of the sites online, I ...
-1
votes
1answer
28 views

Is a monotone function in $[a,b]$ NOT integrable?

I've read many proofs that a monotone function on $[a,b]$ is (Riemann) integrable. If the function is $f(x)=x$ for every rational number and undefined for any non-rational number? I know that the ...
-2
votes
1answer
39 views

Integrating differential forms on curves [closed]

How can I integrate the differential form $$\omega=x\,dx+y\,dy+z\,dz$$ in $\mathbb R^3$ on the curve $$c:[0,2\pi]\to\mathbb R^3: t\mapsto (e^{t\sin t}, t^2-2\pi t, \cos \frac{t}{2})?$$ Some advice ...
0
votes
1answer
29 views

Prove $(\text L)\int_0^1[x-\text K(x)]\sin x\text d x= (\text L)\int_0^1x\sin x \text dx$

Let $\text K(x)$ be a Cantor function on $[0,1]$ prove $$(\text L)\int_0^1[x-\text K(x)]\sin x\text d x= (\text L)\int_0^1x\sin x \text dx$$ here $(\text L)$ denotes Lebesgue-integral. Attempt: ...
2
votes
1answer
48 views

Why $C=1$ in indefinite integral $\int{\sin x\,\mathrm{d}x}$

I am reading Introduction to Calculus and Analysis by Richard Courant. In Section 3.15, g.The Dirichlet Integral, it said $\int{\sin x \,\mathrm{d}x}=1-\cos x$,why $C=1$ here?
-1
votes
2answers
24 views

Linear Integral in complex plane

Integrate using Indefinite Integration and Substitution of Limits Integration symbol with $c$ in denominator $ \mathrm{Re}(z) \, \mathrm{d}z $, $C$ the shortest path from $1 + i$ to $5 +5i$.
4
votes
1answer
42 views

$f_n \to f$ almost everywhere and $\int |f_n| \to \int |f|$ implies $\int |f_n - f| \to 0$?

Suppose $f_n$ and $f$ are integrable, $f_n \to f$ almost everywhere, and $\int |f_n| \to \int |f|$. Does it necessarily follow that$$\int |f_n - f| \to 0?$$
2
votes
1answer
25 views

Integral of sequence converges? [closed]

Suppose $(X, \mathcal A, \mu)$ is a measure space, $f$ and each $f_n$ is integrable and nonnegative, $f_n \to f$ almost everywhere, and $\int_X f_n \to \int _X f$. Does it necessarily follow that for ...
2
votes
1answer
24 views

Sequence of nonnegative $f_n$ tending to $0$ pointwise where $\int f_n \to 0$, but there's no integrable function where $f_n \le g$ for all $n$?

What is an example of a sequence of nonnegative functions $f_n$ tending to $0$ pointwise such that $\int f_n \to 0$, but there is no integrable function such that $f_n \le g$ for all $n$?
2
votes
1answer
15 views

$f$ integrable, if either $A_n \uparrow A$ or $A_n \downarrow A$, then does it follow that $\int_{A_n} f \to \int_A f$? [closed]

Suppose $f$ is integrable. If either $A_n \uparrow A$ or $A_n \downarrow A$, then does it follow that $\int_{A_n} f \to \int_A f$?
1
vote
1answer
56 views

Evaluating $\int_{s^{-1/n}}^{\infty}v^2\exp{\left[-\left(\frac{l}{v} + m v\right)^2\right]}dv$

I am trying to evaluate the following $$I = \int_0^s u^{-3n-1} \exp{\left[-\left(l u^n + \frac{m}{u^{n}}\right)^2\right]}\,du,$$ where $l, m$ and $n$ are positive constants. I tried to substitute $v ...
1
vote
1answer
59 views

Integration by Parts and the Constant of Integration [duplicate]

The constant of integration only seems to be used at the very end of integration by parts despite the use of integrals beforehand. An example of this would be: $$\int x\sin(x)\ dx = x\int sin(x)\ dx -...
0
votes
0answers
28 views

Double Integration in Polar Coordinates

$\iint 2x-y$ $dA$ in the first quadrant and enclosed by $x=0$ $y=x$ and $x^2+y^2=4$ Since the function is enclosed in the first quadrant then $0 \leq \theta \leq \frac{\pi}{2}$ and since $y=x$ and $x=...
-5
votes
2answers
136 views

A new “differential” form for the antiderivative?

The derivative is in general notated by: $\frac {dy}{dx} = \frac d{dx} f(x)$ It has come to my understanding quite recently that dx and dy are actual quantities and not just notational garbage. So ...
1
vote
1answer
48 views

How do I prove this derivation of a definite integral?

Q,How to prove that $\int_{0 }^{\Pi /2}\sin ^{m}x \cos ^{n}x dx =\left [{(m-1)(m-3)(m-5)...2 or 1}\right ]\left [ \left ( n-1)\left ( n-3 \right )..2 or 1 \right ) \right ]\div \left [ \left ( m+n)(...
2
votes
1answer
69 views

For which $\alpha$, $\beta$ does $\int\limits_1^{\infty} x^{\alpha} \cdot (\ln x)^\beta dx$ converge? [duplicate]

For which $\alpha$ and $\beta$ does the following integral converge ?: $$ \int_{1}^{\infty}x^{\alpha}\,\ln^{\beta}\left(x\right)\,\mathrm{d}x $$ Here is my analysis: I noticed that the function ...
0
votes
1answer
40 views

Integration By Parts stuck

Given this S-L problem with solution: And I want to prove this equality when $f(x)$ also has the same conditions as S-L problem. I tried doing integration by parts on the left side twice but ...
1
vote
1answer
61 views

Value of $\int_{-\infty}^\infty\frac{\cos x}{1+x^2} \, dx$

I am a bit puzzled by the expression $\displaystyle I=\int_{-\infty}^\infty\frac{\cos x}{1+x^2}\,dx$. If I try solving it using Cauchy's formula, I arrive to $I=2\pi i \frac{\cos i}{2i} = \pi\cos i$. ...
1
vote
1answer
33 views

prove integral of p(x)q(y) = integral of p(x) * integral of q(y)

I am trying to prove the expression below $$\iint\limits_R p(x) q(y)dxdy = \left( \int_a^bp(x) dx \right) \left( \int_c^dq(y) dy \right)$$ with $R = [a,b] \times [c,d]$ EDIT : $p(x)$ and $q(x)$ ...
1
vote
0answers
39 views

Area of the graph $z=\sqrt{x^2+y^2}$ over the ring $r^2 <(x^2+y^2)<4$ with $0<r<2$

Could someone please help me calculate the surface area of the graph $z=\sqrt{x^2+y^2}$ over the ring $r^2 <(x^2+y^2)<4$ with $0<r<2$ Thanks!
4
votes
1answer
92 views

Fourier Series for $f(x)=\sin(x)+\cos(2x)$

Find the Fourier Series for $$f(x)=\sin(x)+\cos(2x)$$ I got $a_0=0$ which seems correct but I'm struggling with $a_n$ and $b_n$. Here are my attempts: $$\begin{align} a_n&=\frac{1}{2\pi} \...
3
votes
2answers
47 views

L^1 convergence and limsup of convergent sequence

I have to solve this exercise: let $f_n$ be a sequence of positive real function defined on a measure space $(X,M,\mu)$ such that $f_n\in L^1(\mu)$ $\forall n\in \mathbb{N}$ and $f_n$ is convergent in ...
2
votes
4answers
80 views

Help with the integral $\int x\sqrt{\frac{1-x^2}{1+x^2}}dx$

I would like to know what is $$\int x\sqrt{\frac{1-x^2}{1+x^2}}dx.$$ I put $x=\tan(y)$ to get integral of $\displaystyle \int \frac{\sin(y)}{\cos^3(y)}.\sqrt{\cos(2y)}dy$ I don't know whether $\sin(x)...
0
votes
0answers
25 views

any help with $\int p(x)\sqrt{1-\frac{k^2}{q(x)^2}}\ dx$?

I know that probably there is no closed form for this integral, but maybe someone has a good idea to reduce it to something a bit better... I need to evaluate the following integral where $p$ and $q$ ...
1
vote
1answer
51 views

Evaluating $\int_0^s u^{-n-1} \exp{\left[-\left(l u^n + \frac{m}{u^{n}}\right)^2\right]}\,du,$

I am trying to evaluate the following $$I = \int_0^s u^{-n-1} \exp{\left[-\left(l u^n + \frac{m}{u^{n}}\right)^2\right]}\,du,$$ where $l, m$ and $n$ are non-negative constants. I tried to substitute $...
1
vote
1answer
57 views

how do I integrate a modulus function. is it possible?

I am solving a question in which after calculating, my instantaneous velocity is $f(t)=\frac{ab\sin(bt)}{\sqrt{2-2\cos(bt)}}$. So I need to find distance in time $T$. For finding distance I need to ...
2
votes
3answers
50 views

what is the solution of $ \int e^{-3} . x^{-3} dx $?

What is the integration of $ \int e^{-3} . x^{-3} dx $ and how to derive it?
0
votes
0answers
40 views

Limit of an integral involving the floor function

I am trying to obtain an asymptotic expansion of $$\int_t^\infty \lfloor x\rfloor \frac {x}{\sqrt{x^2-t^2}} \ \Bbb d x$$ for $t \to \infty$ and $\lfloor x \rfloor $ denoting the floor function. I ...
0
votes
1answer
38 views

Definite integrals evaluation

I have two results: $$$$ $\int_2^3f(x)dx=5$ and $\int_1^2 xf(x^2-1)dx=8$ I need to calculate: $$\int_0^2 f(x)dx$$ I have no idea about using the previous results, any hint?
0
votes
0answers
31 views

Explanation of “When two functions $a(x)$ and $b(x)$ are not correlated on domain $X$, they can be separately integrated”

Hy everyone, I was reading this paper https://hal.inria.fr/hal-00942452v1/document , and I came up with a statement that I don't fully understand, nor could I found any info on it, so decided to ask ...
5
votes
1answer
77 views

Finding $\int \frac{\mathrm{d}x}{1 + \frac{2}{x} - x}$

I want to solve: $$\int\frac{1}{1+\frac{2}{x}-x} \mathrm{d}x $$ I don't know how to start, maybe I should use partial fraction?