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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1answer
18 views

The volume of a cone whose length of its side is R

How can i compute the volume of a cone whose length of its side is $R$ and the vertex of the cone forms an angle $2 \theta$ . The top cone is a cap of a sphere of radius $R$. Some help please.
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0answers
22 views

How to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$?

Could you please give me some hint how to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$ when f(x) is continuous for $0<x\le1$ and $\lim_{x\to0^+}x^3f^2(x)=1$ ? I tried the usual way: ...
4
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0answers
39 views

Delicate Integral $I=\int_0^\infty \frac{\log^2 x \cos ax}{x^n-1}dx$

Hi I am trying to calculate $$ I:=\int\limits_0^\infty \frac{\log^2 x \cos (ax)}{x^n-1}dx,\quad \Re(n)>1, \, a\in \mathbb{R}. $$ Note if we set a=0 we get a similar integral given by $$ ...
4
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0answers
41 views

Fancy Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$

Hi I am trying to integrate and obtain a closed form result for $$ I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx. $$ Here is what I tried (but I do not think this is ...
2
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1answer
33 views

Integrating Differential Forms

This is part of a homework problem. I want to actually solve it myself, so no solutions, please (although this isn't even the full problem statement). I don't have a very good grasp on differential ...
3
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0answers
25 views

Area interpretation of integrals

When integrating under part of a circle, as in $$A=\int_0^a {\sqrt{r^2-x^2}\,\mathrm{d}x}$$ I noted that the simple geometric solution would be to add the areas of the sector and triangle formed by ...
3
votes
1answer
43 views

Sum as an integral

Recently I have encountered weird notation that I don't see into. When I have some infinite sum $$\sum_{n=1}^{\infty}f(n)$$ I would rewrite it without thinking to the integral form like this ...
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0answers
15 views

Definite integral of greatest integer function

I need to find the area under a function modeled by $f(x)=\left\lfloor 2.4x \right\rfloor+5$. I can't seem to figure out what the antiderivative of this is, so I'm going try to use a right Riemann ...
1
vote
1answer
19 views

Let $\int_a^bf(x)sgn(f(x)) + 2f(x) \ dx = 0$. Show that $f$ has at least one root.

The Assignment: Let $a,b \in\mathbb{R}$ and $a < b$. Furthermore let $f: [a,b] \rightarrow \mathbb{R}$ be differentiable and $|f(x)| + |f'(x)| \neq 0$ for $\forall x \in [a,b]$. Now, let ...
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2answers
26 views

An integral involving two variables and the floor function

Let $N$ be some fixed positive integer. I have the following function $$ g(z) = z \int_1^N [t] e^{2 \pi i t z} \ dt. $$ How would one compute $$ \int_0^1 g(z) \ dz ? $$ Thanks!
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1answer
29 views

Limits of integration

Is there any difference between the $$\int_a^b f(x) dx $$ and $$\lim_{x\to b^-} \int_a^x f(x) dx \qquad \text{OR}\qquad \lim_{x\to a^+} \int_x^b f(x) dx$$ When would one need the second versions of ...
0
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1answer
14 views

Showing that this function is not riemann integrable.

Consider the function h defined by h(x) := x+1 for x an element of [0,1] rational, and h(x) := 0 for x an element of [0,1] irrational. Show that h is not Riemann integrable. The hint in the back of ...
1
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3answers
65 views

Show $\displaystyle\int_0^af(x)g(x)dx\ge\int_0^af(a-x)g(x)dx$

Assume $f$ and $g$ are monotonically increasing on $[0,a]$, Show that $$\displaystyle\int_0^af(x)g(x)dx\ge\int_0^af(a-x)g(x)dx$$ If I differentiate both sides w.r. to $a$ then; ...
2
votes
2answers
19 views

Finding an integral $\int g(x)^j dx $ from $\int g(x)^2 dx $

let $I = \int_0^1 g(x)^2 dx $, where $g$ is a real valued function. With this information is it possible to give an upper bound for $\int_0^1 g(x)^j dx $? Here $j$ is a natural number. When $j=1$ I ...
3
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2answers
45 views

How to show that $f$ is Riemann integrable

Let $u,v:[a,b]\rightarrow\mathbb{R}$ be contunious. Define $f:[a,b]\rightarrow\mathbb{R}$ by $$f(x) = \begin{cases}u(x) & x \in \mathbb{Q} \\ v(x) & x \in \mathbb{R}-\mathbb{Q}\end{cases}$$ ...
2
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0answers
56 views

How to integrate $\displaystyle\int_o^\pi\frac{dx}{\sqrt{3-\cos(x)}}$?

How to integrate $\displaystyle\int_o^\pi\frac{dx}{\sqrt{3-\cos(x)}}$ ? If I take $y=\sin\left(\frac{x}{2}\right)$ then, $\displaystyle ...
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votes
1answer
22 views

Definition of the integral of a vector field on Riemannian manifold and Euclidean spaces

Given a compact Riemannian manifold $(M,g)$ and a vector field $X \in \mathfrak{X}(M)$, is it possible to define the integral of $X$ on $M$? What if $M$ is a Euclidean space? Clearly the definition ...
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0answers
38 views

Turning a summation into an integral

I have a summation of the form: $$y(x) = \sum\limits_{h=-L}^L\frac{A(h)\cdot R(h)^2}{((x-h)^2+R(h)^2)^{3/2}}$$ Where I wish to solve/optimise $R(h)$ (leaving $A(h) = const/h$) or $R(h)$ and $A(h)$ ...
2
votes
2answers
35 views

What is the answer to $\int x(t)dt$?

$\int x(t)dt$? I'm trying to solve a differential equation, but I've hit a strange brick wall that I never used to have a problem climbing over. This question is about mechanics & the equation ...
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0answers
20 views

How to evalute $\displaystyle\int_{2}^{x}\frac{t\ dt}{(\ln^{m} t)(\ln^{n} (t+2))}$

How to evaluate this integral? $$\int_{2}^{x}\frac{t\ dt}{(\ln^{m} t)(\ln^{n} (t+2))}$$ Where $m$ and $n$ are positive integers.
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1answer
11 views

Volume between paraboloid and plane

I need to find the volume of the finite region enclosed between the surface $$ y = 1 - x^2 - 4z^2 $$ and the plane $$y = 0$$ Here's what I've done: $$ \int\int ...
1
vote
1answer
34 views

Integral inequality with first two moments equal to $1$.

Let $f\in \mathcal{C}^0([0,1],\mathbb{R})$ such that $$ \int_0^1 f(x)\text{d}x = \int_0^1 xf(x)\text{d}x=1.$$ Show that $\int_0^1 f(x)^2 \ge 4$. I tried to use Cauchy-Schwartz inequality such that ...
7
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3answers
92 views

Why is $\displaystyle\int^{\infty}_{0}{(1-\cos x)\over{x^{2}}}dx = \frac\pi{2}$?

I have been having trouble understanding Fourier series and analysis in one of my classes. This is one problem from the text and we have to show that this is true. I have done other problems related ...
0
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1answer
34 views

Support of $L^p$ functions?

I noticed something strange. If we look at a function $f \in L^p$, then this is an equivalence class. Hypothetically: $\operatorname{supp}(f) = \overline{\{f\neq 0\}}$. But this is strange, as $f$ is ...
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0answers
92 views

Some users are mind bogglingly skilled at integration. How did they get there?

Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like this. Especially in a ...
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0answers
25 views

Convolving two functions

I'm trying to convolve two functions $f$ and $g$. $$f(x) = e^{-\frac{{(x-p_2)}^2}{2 q_2^2}}$$ $$g(x) = \left(i_1 e^{-\frac{(a-x)^2}{2 \sigma ^2}}+j_1 e^{-\frac{(b-x)^2}{2 \sigma ^2}}\right) \left(i_0 ...
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1answer
26 views

Moments of inertia of a torus

So, I can show that the moment of inertia of a torus about its axis of symmetry is $I_z = 4\pi^2\rho r^5\left[ \frac{3a}{8b} + \frac{a^3}{2b^3}\right]$ where $a$ is the distance from the axis to the ...
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0answers
25 views

Riemann Integrating a Step Function

So I've been trying to prove a step function with countably infinite discontinuities is Riemann integrable using only properties of Riemann integration, no Lebesgue or gauge integration for example. ...
0
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0answers
27 views

Generate Borel Sigma Algebra

I want to show that the Borel Sigma-Algebra on $\mathbb{R}^n$ is generated by $ A:= \{(a_1,b_1] \times \cdots\times (a_n,b_n]; a_i,b_i \in \mathbb{R} \}$ as well as $ B:= \{(-\infty,c_1] ...
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0answers
25 views

change of variable in the integral with measure $\mu$ [on hold]

Let $f:(X,\mathbb{X}, \mu) \to Y$ a measurable function and $\nu(A)=\mu(f^{-1}(A))$. show that $$\int_Yg \,d\nu = \int_X g(f(x)) \,d\mu(x)$$
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votes
2answers
49 views

Calculating a double integral

Calculate $$\int_{D}(x-2y)^2\sin(x+2y)\,dx\,dy$$ where $D$ is a triangle with vertices in $(0,0), (2\pi,0),(0,\pi)$. I've tried using the substitution $g(u,v)=(2\pi u, \pi v)$ to make it a BIT ...
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2answers
59 views

Show the function is integrable and find the integral - somewhat complex question

We are given $Q = [0,1]$x$[0,1]$ We are also given the function $f(x,y) = (\frac{1}{10})^n$ where $\frac{1}{2^{n+1}} < \max(x,y) \leq \frac{1}{2^n}, (n=0,1,2,...)$ and $f(0,0)=0$. Show that $f$ ...
2
votes
4answers
72 views

Evaluating the improper integral $\int_0^{\infty} \frac{\sin x}{x+x^2} \ dx$

Evaluating the improper integral $$\int_0^{\infty} \frac{\sin x}{x+x^2} \ dx$$ I'm trying to determine if the integral exists. I can't seem to deal with $$\lim_{a\to 0^+} \int_a^\infty \frac{\sin ...
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1answer
27 views

I need to show that this sequence is increasing and I'm almost there but I need help on last step.

Let $(1+\frac{1}{n})^n$ be a sequence and $f(x)=(1+\frac{1}{x})^x $ on $[1,inf)$. I need to show that f is non-decreasing by showing that $f'(x)\ge0$. So far I have: Let $g(x)=ln(f(x))$, where $ln$ ...
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0answers
13 views

Mehler-Sonine integrals for $Y_{\nu}(x)$ Bessel function

My question is regarding the following integral representation of the $Y_{\nu}(x)$ Bessel function (Watson page 170): $$ Y_{\nu}(x) = \frac{2 (\frac{1}{2} x)^{-\nu}}{{\pi}^{1/2} \Gamma(\frac{1}{2} - ...
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2answers
29 views

Calculus power series

Hi could anyone help me to solve this. express the function $\int_x ^0 (\sin(t^2)\cdot \cos(t^2))$ as a power series. Because there is two trigo identies I do not know how to combine them to form a ...
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1answer
39 views

L1 convergence and Lp bounded implies Lq convergence

I have tried to solve this problem for almost a week and did not manage to, so I figured to ask it here: Let $(u_n)\to u$ in $L^1(0,1)$ strongly and let $\{u_n\}_{n\in\mathbb{N}}$ be bounded in ...
0
votes
1answer
29 views

Show that for $|f(z)| \leq C (|z| + 1)\log(|z| + 1)$, there is an $a$ such that $f(z) = az$

Let $f: \mathbb{C} \to \mathbb{C}$ be analytic and suppose a $C \geq 0$ exists such that \begin{align*} |f(z)| \leq C(|z| + 1) \log(|z| + 1) \end{align*} for all $z \in \mathbb{C}$, where $\log: ...
3
votes
2answers
49 views

a geometric interpretation of a line integral

Is there a geometric interpretation of the line integrals : $\int_{\gamma} f(x,y)\, dx$ $\int_{\gamma} f(x,y)\, dy$ (which should not be confused with $\int_{\gamma} f(x,y)\, ds$) where ...
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1answer
39 views

A question about improper integral

Could you please give me some hint how to solve this problem: Suppose f(x) continuous in $[0,\infty)$ and for each a,b>0 and c>b $ab \left|\int_0^1 f\left(ax+c \right) dx \right|<1$. Prove ...
2
votes
2answers
40 views

Computing $\int_{\gamma} {dz \over (z-3)(z)}$

Compute, using the Cauchy Integral Formula, $$ \int_{\gamma} {dz \over (z-3)(z)} $$ where $\gamma$ is the circle of radius $2$ centered at the origin, oriented counterclockwise. ...
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0answers
33 views

Line integral - should I parametrize the square?

I have the following $1-\text{form}$ defined: $$\omega = \displaystyle\frac{2xy}{(1-x^2)^2+y^2}\mathrm{dx}+\displaystyle\frac{1-x^2}{(1-x^2)^2+y^2}\mathrm{dy}$$ I'd like to find ...
4
votes
1answer
103 views

Show $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $

Need to prove $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $ and $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{y}{1+y^2}e^{-ay} dy =0 $ Can ...
1
vote
2answers
28 views

Triple integral problem involving definite integrals (and Taylor's formula possibly)

Any hint on how to approach this question? Show that $$\int_0^x\int_0^v\int_0^u f(t)dtdudv=\frac12\int_0^x(x-t)^2f(t)dt$$ I am completely clueless. I tried to convert it into the standard xyz form ...
10
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1answer
232 views

Cool Integral = $\pi/2$ !!

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}dx. $$ (I have literature on this, if people want). Note, we can write the ...
0
votes
1answer
20 views

Bartle - integration, monotone convergence theorem

Suppose that $(f_n) \subset M^{+}(X, \mathbb{X})$, that $(f_n)$ converges to $f$, and that $\int f d\mu=\lim \int f_n d\mu < +\infty$. Prove that $$\int_E f d\mu=\lim \int_E f_n d\mu $$ for each ...
3
votes
1answer
33 views

Properties of a Mehler's type integral

When computing the resolvent of the Laplace beltrami opetator on $S^n$ for even dimension, $n=2k$, I came across the following integral $$ ...
4
votes
1answer
68 views

Asymptotics of an oscillatory integral with a linear oscillator

I am interested in asymptotic results for $$ S(p) = \int_0^1 \frac{y \sqrt{1-y^2}}{(\varepsilon^2-1)y^2+1} \sin(py) dy, $$ i.e. a result that is valid as $p\rightarrow\infty$. The parameter ...
0
votes
3answers
29 views

Integrating the product of Poisson and exponential pdf

So I'll spare the background as to why, but I'm trying to integrate the following: $$\int_0^{\infty} \frac{e^{-(\lambda+\mu)t}(\lambda t)^n}{n!} dt$$ If you parameterize a Poisson w/ $\lambda$ and ...