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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9 views

Integrating to find deceleration, and finding ball height?

1) A ball is thrown straight up from a height of 8 ft with an initial velocity of 40 ft/sec. How high will the ball go? (Take g = 32 ft/sec2.) How would I do this? Wouldn't I need to find a velocity ...
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0answers
10 views

Given $f\in L^1(\mathbb{R})$ with $\|f\|_1<\infty$ and $g_n=\sqrt{n/2\pi}e^{-nx^2/2},f_n=g_n\ast f$, show that $\lim\|f_n-f\|_1=0$

Given $f$ a Lebesgue integrable function on $\mathbb{R}$ with finite $L^1$-norm, I am asked to show that $\lim_{n\to\infty} \|f_n - f\|_1 = 0$, where $f_n = f \ast g_n$ and $g_n = ...
2
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1answer
34 views

Showing that $\int fg\le \int g$ implies $f=0$ a.e.

Take $0<p<1$. If $f$ is locally integrable over on $\mathbb{R}$ and $$\Bigg\vert \int fg\Bigg\vert\le \Vert g\Vert_p\tag 1$$ for every $g$ continuous on a set of compact support, then $f=0$ a.e. ...
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6 views

The $L^p$ convergence rate of the tail of the series $\sum_{n=1}^{\infty}\min\{1,2^n |x|^{-1} \}2^{-na}$

This a follow-up to the question: Convergence Rate of the Tail of the Series $m^{a}\sum_{n=1}^{\infty}\min\{1,2^n m^{-1} \}2^{-ja}$ When $a > 0$, we have $$ \sum_{n=1}^{\infty}\min\{1,2^n |x|^{-1} ...
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7 views

Lower semicontinuous non-negative function on a locally compact Hausdroff space with a countable base

An extended real number is an element of $\mathbb R \cup \{-\infty, +\infty\}$. Let $X$ be a locally compact Hausdorff space with a countable base. An extended real valued function $f$ on $X$ is ...
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0answers
27 views

Any hint for solving this Poisson's integral?

I tried various approach without success in solving this integral: $\frac{1}{2\sqrt{\pi t}}\int_{\mathbb{R}} e^{\frac{-(x-y)^2}{4t}}\phi (y) dy$ which is the solution to the heat equation. I only have ...
2
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1answer
35 views

Integration over ellipse

$A=\{(x,y)\in \Bbb R^2\mid \frac{x^2}{a^2}+\frac {y^2}{b^2}=1\}$. Find $\int_A (\cos x)y\,dx+(x+\sin x)\,dy$. Can someone please please give a methodological answer? Thanks a lot!
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15 views

A question of multi-dimensional integral

Consider the function $$\Omega(N,E)=\int dE_1 \int dE_2 \cdots \int dE_N \Omega_1(E_1)\Omega_2(E_2) \cdots \Omega_N(E_N)\delta(E-E_1-E_2\cdots -E_N)$$ Is there a sufficiently condition on the ...
2
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1answer
63 views

Bonus integration problem we got at class: Integrate $\frac {x \sin x}{1+\cos^2x}$ between $0$ and $\pi$ [duplicate]

Bonus integration problem we got at class: Integrate $\frac {x \sin x}{1+\cos^2x}$ between $0$ and $\pi$ So the lecturer gave this problem. I tried this really hard but couldn't succeed. It ...
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3answers
69 views

Integrate $\frac{1}{1+\cos^2x}$. Probably need using some trigonometric identity I don't know

Integrate $\frac{1}{1+\cos^2x}$ I probably need using some trigonometric identity I don't know. I tried all methods I'm familiar with. Any assistance will be great. Thank you!
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2answers
37 views

How does the first fundamental theorem of calculus guarantee the existence of antiderivatives of functions?

First fundamental theorem of calculus: $$g(x) = \int_a^xf(t)dt$$ then $$g'(x) = f(x)$$ But how does this guarantee the existence of antiderivatives of functions? Tutorials always state it does, but ...
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3answers
52 views

Integral of trig fraction using substitution?

I'm chewing on an integral problem and don't have a clue where to begin. If someone could assist by suggesting a good starting point, I'd really appreciate it! Not asking for anyone to solve the ...
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2answers
28 views

Evaluating a complex integral (Hints please)

I am supposed to be able to show that, given $f(z)=\frac{1}{\pi}\int_0^1r\int_{-\pi}^\pi\frac{d\theta}{re^{i\theta}+z}dr$ then $f(z)=\overline{z}$ for $|z|<1$ and $f(z)=1/z$ if $|z|\geq1$. (This ...
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0answers
33 views

Integration of a function of two variables

How can we check the integrability of $f$ defined on $[0,1] \times [0,1]$ as $f(x,y)=$\begin{cases} 0 & x=\frac{1}{2},y \in \mathbb Q \\ 1 & x=\frac{1}{2},y \in \mathbb Q^c \\ ...
2
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0answers
26 views

If $A$ and $f$ are bounded, then $f$ is integrable in the extended sense (?) [Spivak]

I have a problem with one of the theorems in Spivak's Calculus on Manifolds. I will give some background first: An open cover $\mathcal{O}$ of an open set $A \subset \mathbb{R}^n$ is admissible if ...
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1answer
23 views

Joint density function problem

I have a joint density function of Random Variables X and Y given by: $$ f(x,y) = \begin{cases} 2e^{-x}e^{-2y} & 0<x<\infty, 0<y<\infty \\ 0 &\text{otherwise} \end{cases} $$ And ...
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0answers
13 views

Correctly setting up flux integrals

My question has to do with picking the correct limits for integration. I thought I had it figured out well, but I had an interesting issue with a homework problem. The problems were about Green's ...
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0answers
29 views

If $f_n \to f$ pointwise a.e., $\int |f| < \infty$, and if $\int |f_n| \to A$, is $A=\int |f|$?

We work on some domain $\Omega$ which may or may not be bounded. If $f_n \to f$ pointwise a.e., if $\int |f| < \infty$, and if we know that $\int |f_n| \to A$ to some number $A$, is $$A=\int ...
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1answer
23 views

How do you integrate $\int _{0}^t\:\dot p(s) p(s) + p^2(s)ds$

Given $p(s)$ some single valued function How can I show that $$\int _{0}^t\:\dot p(s) p(s) + p^2(s)ds$$ has resulting in something along the line of $$\frac{p^2(s)}{2}$$ note $\dot p(s)$ signifies ...
3
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3answers
215 views

Integral involving Bessel functions of the first kind

I am stuck with the following integral. Does it converge? $$ \int_{0}^{\infty}\left(J_1(x)^2+J_1(x)J_1(x)^{''}\right)\text{d}x $$ According to tables I find that the first term is divergent, so I ...
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3answers
52 views

How to find $p(t)$ when $m$ varies linearly with $t$? [on hold]

I have a function $p(t)$ (position and time) defined by $$p(t) = \frac{1}{2} \cdot \frac{F}{m} \cdot t^2$$ when the mass is constant. This is derived from Newtons second law and by integration of the ...
2
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4answers
95 views

Compute $\int _{\frac{4}{5}}^2\:f^{-1}\left(x\right)dx$

We have to compute $\int _{\frac{4}{5}}^2\:f^{-1}\left(x\right)dx$ where $f\left(x\right)=\frac{-x^3+2x^2-5x+8}{x^2+4},\:x\in \mathbb{R}$ is an bijective function. How help if we kno![enter image ...
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0answers
28 views

Dirac delta question from “Classical covariant fields” by Burgess

If you have the book with you. Kindly tell me how did he reach equation 2.54 from equation 2.52. I tried to solve the delta function according to given instruction but I am making some mistake. Kindly ...
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2answers
87 views

How to solve $\int \frac{(x-1)\sqrt{x^4+2x^3-x^2+2x+1}}{x^2(x+1)}dx$?

I need to compute $$\int \frac{(x-1)\sqrt{x^4+2x^3-x^2+2x+1}}{x^2(x+1)}\ dx.$$ I tried it on wolfram but it timed out, maybe because I am on a mobile device. Any hint is appreciated.
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0answers
75 views

Can we integrate less than 0.00001% of functions? [on hold]

I'm told that we can integrate less than 0.00001% of functions. Is this true? Any proof?
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0answers
14 views

Characteristic function of an asymmetric Laplace distributed random variable

What is the characteristic function of a random variable with density $$f_X(x) = \frac{1}{2} [ 1_{x>0} \, a e^{-a x} + 1_{x<0} \, b e^{b x} ], \; \; \; \quad a,b > 0 \quad \quad ? $$ My ...
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3answers
147 views

How to solve this integral by a simple way?

I'm given $$\int \frac{x^3}{\sqrt{x^4+x^2+1}}dx$$ My attempt, Let $u=x^2$, $du=2xdx$ $$=\frac{1}{2}\int \frac{u}{\sqrt{u^2+u+1}}du = \frac{1}{2}\int ...
3
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1answer
28 views

Is there a spherical coordinates system for vectors of complex numbers?

Suppose I have a scalar field $f(\vec{x})$, where $\vec{x}\in\mathbb{R}_3$, and I wish to average $f$ over a sphere $|\vec{x}|=R$: $\displaystyle\langle f\rangle_{R} = \frac{\int_{S} f(\vec{x})\, ...
1
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1answer
48 views

Integral of ln (3x) / x

I believe this should be a simple problem but I don't have an answer key to confirm if this is right, and some of the similar questions I can find online seem to be giving more complicated solutions. ...
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3answers
82 views

$f(x)$ is Riemann integrable $\Rightarrow$ $\frac{1}{1 + f^2(x)}$ is Riemann integrable

Let f(x) be Riemann integrable on [a,b]. Then there exist $\lim_{x \rightarrow a+0} f(x)$ and $\lim_{x \rightarrow b-0} f(x)$ f(x) has only removable or jump discontinuities. The set of ...
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0answers
17 views

Predicting equality/inequality of integrals of multivariable functions

Is it possible to predict equality/inequality, of indefinite integrals of multivariable fucntions, over a domain from equality/inequality respectively of those functions over the same domain? Does ...
1
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1answer
22 views

The value of x satisfying $\int^{2[x+14]}_0\{\frac{x}{2}\}dx =\int^{\{x\}}_0[x+14]dx $ where [.] …

Problem : The value of x satisfying $\int^{2[x+14]}_0\{\frac{x}{2}\}dx =\int^{\{x\}}_0[x+14]dx $ where [.] denotes the greatest integer function and $\{.\}$ denotes the fractional part function. ...
1
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1answer
20 views

Conditions on $f(t)$ so that $\int_{-\infty}^\infty f(t) \operatorname{sinc}(t-a) \operatorname{sinc}(t-b) dt$ converges.

Let us consider $$\int_{-\infty}^\infty f(t) \operatorname{sinc}(t-a) \operatorname{sinc}(t-b) dt \ \ \ \ (*)$$ for $a,b\in \mathbb R$. If $f\in L^1(-\infty,\infty)$ the integral converges: ...
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0answers
17 views

Change of variables in double integration

I was trying to solve this double integral $\int_{0}^{1}\int_{0}^{y}(1-x)^{59}(y-x)^{27}dxdy$, I could do this by taking binomial expansion but that would be very painful. So a sure thing here is a ...
1
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1answer
20 views

Given $f\in L^1(\mathbb{R})$ with $||f||_1 < \infty$, is it true that $\int_{\mathbb{R}} ||f||_1 - f(x) \, dx = 0$?

According to my intuition so far, the answer should be yes, hinging very important on the assumption that $||f||_1 < \infty$. To speak very roughly, if the $L^1$ norm of $f$ is finite, it seems ...
1
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1answer
36 views

Triple integral volume by equations

I have trouble setting up a triple integral to find volume bound by equations, such as: $$z = x^2 + 3;\quad y = 3 - x^2;\quad x + y = 2;\quad z = 0.$$ I'm not sure how to figure how to find the ...
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1answer
32 views

Find the volume of the region bounded by $ (x^{2}+y^{2}+z^{2})^{2}=x$

I tried to convert it to spherical coordinates to find the bounds: $(p^{2})^{2} = p\sin(\phi) \cos(\theta)$ => $ p^{3} = \sin(\phi)\cos(\theta)$ not sure where to go from here.. $ 0 < \theta ...
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1answer
54 views

Foundation calculus doubt

So I have an ODE in the following form: $\frac{dx}{d\text{t}} = f(\text{m}) sin\text{z}$ where z = z(t) and m = m(t) i.e. they are both functions of time, t. Now, if I were to concern It is possible ...
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1answer
29 views

On the Preservation of Product Measurability under Partial Conditional Expectation.

Let $(X,\mathcal{X},\mu)$ and $(Y,\mathcal{Y},\nu)$ be probability spaces, $\mathcal{X}_{0}\subset\mathcal{X}$ a (sub)sigma field and assume that $f=f(x,y)\in L^{1}_{\mu\otimes \nu}$ where $(X\times ...
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2answers
36 views

Volume of solid by Spherical

Trouble setting up the integrals for this problem. Find the volume of the solid bounded by $x^2 + y^2 = 1, z = 0$, $z = 6$, $y\geq 1/2$. Use integration with Spherical coordinates. (Hint: Use two ...
2
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3answers
40 views

How to prove and evaluate an Improper Integral

How to show that this improper integral converges and how to compute its value? $$ I=\int_{0}^{\frac\pi 2}\frac{\cos(2t)}{\sqrt{\sin(2t)}}\mathrm{d}t. $$ I used that the integrated function is odd so ...
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2answers
78 views

Clever way of finding $\int_0^\infty x\Phi(x)\phi(x)dx$

Suppose that $\Phi$ and $\phi$ are the Standard Normal c.d.f and p.d.f. respectively. Then, evaluate $$\int_0^\infty x\Phi(x)\phi(x)dx$$ There is no use of my trying to show my approach because ...
0
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1answer
35 views

Difficult integration

In my notes the lecturer takes the Fourier transform in $x, y$ and $t$ of $\phi(x,y,z,t)$ as: $$ \int_{-\infty}^{\infty}dt\, e^{i\omega ...
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3answers
105 views

How to evaluate $\lim _{x\to 0}\frac{1}{x^5}\cdot \int _0^x\:f\left(t\right)dt$ with MVT

We have to evaluate $\lim _{x\to 0 }\frac{1}{x^5}\cdot \int _0^x\:f\left(t\right)dt$ where $f\left(x\right)=x-\frac{x^2}{2}+\frac{x^3}{3}-\log\left(1+x\right)$ , $\:x\in \left(-1,\infty \right)$ I ...
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1answer
37 views

Triple Integrals

Find the volume of the region bounded by $(x^2+y^2+z^2)^2=x.$ I'm having issues setting my bounds (specifically $\theta$) so far I have $0<r< [\sin(\phi) \cos(\theta)] ^{(\frac{1}{3})}$ ...
2
votes
2answers
33 views

Showing that supremum function is integrable

Let $g_1(\omega),g_2(\omega),...$ be integrable functions defined on $\Omega$ with $g_n\rightarrow g$ and $g$ is integrable and also $\lim \int g_n=\int g$ . Define $h(\omega)= \sup_n g_n(\omega)$. ...
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0answers
36 views

Is there a closed form expression for the following definite integrals?

I am looking for a closed form for these two integrals $$\int_{-\infty}^{-a}\text{d}x \frac{1}{|x|}e^{-\frac{1}{2}x^2\sigma^2}e^{i k |x|}+\int_a^{\infty}\text{d}x ...
1
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1answer
30 views

Existence of double integral

the short time Fourier transform is obtained by the formula: $$Sf(u,\epsilon)=\int_\mathbb{R}f(t)g(t-u)e^{-i\epsilon t}dt$$ where $f,g \in L^2(\mathbb{R})$ are the signal and window respectively: ...
1
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0answers
22 views

How to partially differentiate an integral with a density function?

I am given this result: $$\frac{\partial}{\partial x(t)} \left[\lambda \int u(x(t)) f(t) \mathrm{d}t\right] = \lambda u^\prime(x(t)) f(t)$$ Where $\lambda$ is a constant, and we have the probability ...
2
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0answers
25 views

Implementing the Risch algorithm to integrate $\dfrac{\log(x)+2}{x^{2}\log^{3}(x)}$

Following the work of Andreas Wurfl i am trying to implement the Risch algorithm on $\int{\dfrac{\log(x)+2}{x^{2}\log^{3}(x)}dx}$ following his method for extensions that are purely logarithmic, we ...