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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1answer
8 views

Integral by using substitution (How to proceed?)

Using the substitution $x=a\sin\theta$, or otherwise, find $\int\frac{1}{x^2\sqrt{a^2-x^2}}d\theta$. My attempt, $x=a\sin\theta$ $dx=a\cos (\theta)d\theta$. Then $\sqrt{a^2-x^2}=\sqrt{a^2-a^2\sin ...
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0answers
22 views

Integral of the reciprocal of the natural logarithm [on hold]

What is the value of this integral $$\displaystyle\int\frac{1}{\ln(x)}\;dx$$
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0answers
6 views

Euler method global truncation error and conservation of orbital energy.

I've been given a simplified model of a small mass orbiting a much larger one, which I need to solve using Euler's method. I've reduced the equation to two (or four, really) coupled first order ODEs: ...
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0answers
8 views

if $F(s_{0})$ for some $s_{o}$exists then it exists for all $s>s_{o}$

if laplace transform $F(s_{0})$ for some $s_{o}$exists then it exists for all $s>s_{o}$. i need to prove this . now, ...
0
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1answer
13 views

Determining Line Integrals from a Graph and Vector Field (Image Included)

Consider the vector field: $$F=(\frac{2xy-2xy^2}{(1+x^2)^2}+\frac{8}{13})i+(\frac{2y-1}{1+x^2}+2y)j$$ Determine $\int_cF*dr$, where $C$ is the path $C_1+C_2+C_3$ from $(2,0)$ to $(5,6)$ shown. I ...
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0answers
30 views

Diffrental equation solution [on hold]

How can I solve this equation? $$\frac{\partial f}{\partial x} =\frac{a-x}{y} \frac{\partial f}{\partial y}$$ where $a$ is a constant. So what is $f$?
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1answer
40 views

Integral of a total derivative

I have seen the "total differential" $$ d \ln A = -d \ln B/c $$ Representing how infinitesimal changes in $A$ are related to infinitesimal changes in $B$. Someone then took the integral of this ...
3
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1answer
25 views

Volume of a Solid of Revolution Rotated Around the Y-Axis

Sorry to post an obvious homework question here, but my daughter's calculus teacher isn't much on "teaching" and left a problem like this one out of the notes. I can't find much on the internet to ...
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2answers
47 views

Antiderivative of $ (x^2 + c)^{-3/2} $ [on hold]

What method should be used to determine the antiderivative of this expression? Edit: I have $ c > 0 $ in the problem I'm working on.
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2answers
23 views

Integrating to find deceleration, and finding ball height? [on hold]

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 ...
3
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1answer
34 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
66 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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0answers
13 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} ...
0
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1answer
9 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
28 views

Any hint for solving this Poisson's integral? [on hold]

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
votes
1answer
37 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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0answers
17 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
71 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
71 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
39 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 ...
3
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3answers
58 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
35 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
14 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
30 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
216 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 ...
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4answers
97 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
31 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 ...
1
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3answers
149 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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0answers
29 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})\, ...
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1answer
49 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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2answers
84 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 ...
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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
19 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 ...
1
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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 ...
0
votes
1answer
56 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 ...
0
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1answer
30 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 ...
-1
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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 ...