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Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

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18 views

Find the volume of the solid bounced by the planes $z=0$,$z=y$ and $x^2+y^2=1$

So I do the following: $$\int_{-1}^{1}\int_0^{\sqrt{1-x^2}} \int_0^{y} \,dzdydx$$, but the answer gives me $\frac{2}{3}$, as it graphs a cylinder it should be the half of the half of a cylinder of ...
0
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1answer
10 views

How to prove the integral formulae of the inverse path $\alpha^-$ and the product path $\alpha\beta$?

I need help with this problem: Let $f:S\subset\mathbb{R}^n\rightarrow\mathbb{R}$ be continuous on $S$, and let $\alpha:[a,b]\subset\mathbb{R}\rightarrow\mathbb{R}^n$ and $\beta:[c,d]\subset\mathbb{...
1
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1answer
31 views

Conditions under which a known vector valued function the gradient of some function

Suppose that we have a vector valued function $D(x)$ with derivative $H(x)$ and that both of these are smooth. Under what conditions does there exist a function $f(x)$ such that $\nabla f(x) = D(x)$? ...
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0answers
18 views

Substitute $g(x)=Ae^{-\beta x^2}$ into $\theta(x,t)=\frac{1}{2\sqrt{\pi\alpha t}}\int_{-\infty}^{\infty} g(\eta)\exp(-(x-\eta)^2/4\alpha t) \ d\eta$

I am trying to show that by directly substituting $g(x)=Ae^{-\beta x^2}$ into $$\theta(x,t)=\frac{1}{2\sqrt{\pi\alpha t}}\int_{-\infty}^{\infty} g(\eta)\exp(-(x-\eta)^2/4\alpha t) \ d\eta,$$ we obtain ...
1
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1answer
46 views

How do I integrate $4\int_0^{\pi/2}\dfrac{\sec^2(\theta)}{1+2\tan^2(\theta)}\,d\theta$ using symmetry?

\begin{align}\int_{-\pi}^\pi \frac{1}{1+\sin^2(\theta)}\,d\theta&=4\int_0^{\pi/2} \frac{1}{1+\sin^2(\theta)}\,d\theta\\\\&=4\int_0^{\pi/2}\frac{\sec^2(\theta)}{1+2\tan^2(\theta)}\,d\theta\\\\&...
1
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1answer
39 views

inequality about $I=\int_{2012}^{3012}\sqrt[3]{x}\,dx$

Consider $$ \begin{align} & L=\sqrt[3]{2012}+\sqrt[3]{2013}+\ldots +\sqrt[3]{3011} \\ & R=\sqrt[3]{2013}+\sqrt[3]{2014}+\ldots +\sqrt[3]{3012} \\ \end{align}\ $$ and $$ I=\int_{2012}...
3
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0answers
50 views

How do you solve this $\int\limits ^{\infty }_{0}\frac{\cos( x)}{x^{n} +1} dx,\ n >0$

Me and my friend have tried a wedge,a triangle, and we even tried Feynman's technique. None of these things got us an answer to the integral $\int\limits ^{\infty }_{0}\frac{\cos( x)}{x^{n} +1} dx,\ n ...
0
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1answer
19 views

Maximum of $\int_0^1(x^p|f(x)|^q-x^q|f(x)|^p)dx$

Let $p>q>0$ and $C=\{f:[0,1] \to \mathbb{R} \mid f \text{ is continuous} \}$. Determine $$\max_{f \in C}\int_0^1(x^p|f(x)|^q-x^q|f(x)|^p)dx$$ and the functions for which this maximum occurs. ...
7
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0answers
66 views

Integral $\int_0^1 \frac{\ln(1+x)}{1+x^3}dx$

Earlier today I saw this integral around here and gave it a try without success, unfortunately it got taken down so it didn't receive to much attention, but I think it's a nice integral (although it ...
0
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1answer
27 views

Calculate initial velocity based on displacement, time and constant acceleration.

"A car has a constant speed along a road. It goes down a hill at a constant acceleration. 50s after it goes down the hill the speed is doubled and 50s later it reaches the end of the 200m hill and is ...
0
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0answers
15 views

Evaluate $\int_{-\infty}^\infty \frac{e^{isx} ds}{(s-\tfrac{7i}{2})^2 + (1/2)^2}$

Evaluate $$\int_{-\infty}^\infty \frac{e^{isx} ds}{(s-\tfrac{7i}{2})^2 + (1/2)^2}$$ Here I am having trouble as using simply the fourier inverse of $\frac{2a}{s^2+a^2}$ which is $e^{-ax}$ gives wrong ...
0
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0answers
8 views

Surface integral of a scalar function in the first octant

I am having difficulty drawing and parameterizing the surface for the integral $$ \iint_s x^2z\,dx\,dy\ $$ s: 1st octant part of $y=x^2\ $ cut by $2x+y+z=1$
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1answer
71 views

Why do I get an error when evaluating $\int_{-\pi}^{\pi} \frac{d\theta}{1+\sin^{2}\theta}$?

Using the trigonometric identity: $$\cos 2\theta = 1 + \sin^{2} \theta$$ I get that $$\int_{-\pi}^{\pi} \dfrac{d\theta}{\cos 2\theta}$$ Then using a property of integrals, I get: $$\int_{-\pi}^{0}\...
2
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1answer
71 views

Prove that $\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$ when $f(f(x))=x^2$

Let $f:[0, \infty) \to [0,\infty)$ be a differentiable function with $f'$ continuous. If $f(f(x))=x^2$, prove that $$\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$$ without explicitly finding $f.$ Since we ...
1
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0answers
27 views

How to evaluate the line integral (checking Stokes Theorem)

Consider the vector field: $$\vec F = ye^x \hat i + (x^2 + e^x) \hat j + z^2e^z \hat k$$ A closed curve $C$ lies in the plane $x + y + z = 3$, oriented counterclockwise. The parametric ...
0
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1answer
23 views

What is a good estimate for $\sum_{t=0}^T \frac{1}{\gamma^t}\frac{1}{\sqrt{t + 1}}$, where $0 < \gamma < 1$?

Let $0 < \gamma < 1$ and $T$ be a "large" nonnegative integer. Question What is a good estimate (upper bound) for $\sum_{t=0}^T\dfrac{1}{\gamma^t}\dfrac{1}{\sqrt{t + 1}}$? In general, for $\...
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0answers
15 views

Solving a Stratonovich SDE

I am trying to solve the following Stratonovich SDE $$dN_t=rN_tdt+\gamma N_t\circ dB_t$$ In my notes, the Stratonovich integral is defined as $$\int^t_0 N_s\circ dB_s=\int^t_0 N_sdB_s+\frac{1}{2}\...
1
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1answer
55 views

Calculate $\int_{\pi\over 2}^{{\pi\over2}+i} \cos2 z dz$

Calculate $\int_{\pi\over 2}^{{\pi\over2}+i} \cos 2z dz$. I want to verify my answer please. My solution: Because $\cos 2z$ is analytic everywhere, we just need to calculate the integral: $$ \...
0
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1answer
42 views

Why defined integral of x works

I know this is duplicate, but I still struggling to understand why this particular example works: If we check integral of x dx, we see it is parabola. https://www....
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0answers
25 views

When do derivatives cancel inside integrals when working with tensors?

While doing a problem recently I realised I'm not clear about when derivatives inside integrals will cancel when working with tensors. For example, I have come across integrals such as: $\int \...
0
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0answers
33 views

How to evaluate the following improper integral? [duplicate]

$\int_{0}^{\infty}\frac{1}{1+x^{2n}}dx = ? $ Using the Beta function and Euler's reflection formula the result of the integral ought to be $\frac{\frac{\pi}{2n}}{\sin{\frac{\pi}{2n}}}$ But I ...
1
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1answer
25 views

Proving $\int_{I} f \geq 0$ for an integrable function that is positive at rationals

Let $I$ be a generalized rectangle in $\mathbb{R}^{n}$ and suppose $f :I \rightarrow \mathbb{R}$ is integrable. Suppose $f(x) \geq 0$ if $x$ is a point in $I$ with a rational component. Prove $\int_{\...
0
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1answer
22 views

Monotonicity of improper integrals [on hold]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$, $g:\mathbb{R} \rightarrow \mathbb{R}$, $x \mapsto f(x)$, and $x \mapsto g(x)$. If $f(x) \leq g(x)$ $\forall$ $x \in \mathbb{R}$, and $\int_{-\infty}^{\infty}...
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3answers
69 views

Is it possible to integrate $\int_0^1 \frac{x^{n}}{x+5} dx + \int_0^1 \frac{x^{n-1}}{x+5} dx$? If so, how?

Maybe there is some method I am forgetting, but I do not remember how to solve this (if it is even solvable). $$\int_0^1 \frac{x^{n}}{x+5} dx + \int_0^1 \frac{x^{n-1}}{x+5} dx$$
0
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1answer
40 views

How to find $f(x)$ if $\int_0^{x^2} (1+t)f'(t)dt=x^4$ and $\int_0^1 f(t)dt=3 $

Knowing that $f:[0,+\infty)\rightarrow \mathbb{R}$ is continuous and derivable, and that: $\int_0^{x^2} (1+t)f'(t)dt=x^4$; $\int_0^1 f(t)dt=3 $ Determine $f(x)$. (Note: This is supposed ...
1
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1answer
20 views

Change of double integral limits

I need to find $\iint_{D}e^{-4x^2-y^2}dxdy$, where $D = \{(x,y) \mid 4x^2 + y^2 \leq 2 \}$. I have to use the change of variables: $x=ucos(v), y=-2usin(v)$. I am able to find the new integrand, but ...
4
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1answer
79 views

Find function $f(x)$ satisfying $\int_{0}^{\infty} \frac{f(x)}{1+e^{nx}}dx=0$

I am looking for a non-trivial function $f(x)\in L_2(0,\infty)$ independent of the parameter $n$ (a natural number) satisfying the following integral equation: $$\displaystyle\int_{0}^{\infty} \frac{f(...
10
votes
3answers
140 views

Evaluate the definite integral $\int^{\infty }_{0}\frac{x \,dx}{e^{x} -1}$ using contour integration

My friend and I have been trying weeks to evaluate the integral $$\int^{\infty }_{0}\frac{x \,dx}{e^{x} -1} .$$ We have together tried 23 contours, and all have failed. We already know how to ...
0
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0answers
26 views

A question regarding inequality between weighted averages

Suppose you have three probability density functions $p_{1}(x)$, $p_{2}(x)$, and $p_{3}(x)$. Suppose further that the expectation values of $p_{1}(x)$ and $p_{2}(x)$ are unequal $\int_{0}^{1}p_{1}(x) ...
0
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0answers
18 views

$-\frac{\rho_0}{4\pi\epsilon_0}\int\limits_{0}^{2\pi}\int\limits_{a}^{2a}\int\limits_{0}^{\frac{\pi}{4}}(\cos{\varphi}\sin{\theta}+\sin{\varphi}…$

So I have the following integral that pops up when trying to solve a physics problem: \begin{equation} -\frac{\rho_0}{4\pi\epsilon_0}\int\limits_{0}^{2\pi}\int\limits_{a}^{2a}\int\limits_{0}^{\frac{\...
1
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2answers
24 views

Incorrect sign when evaluating of bounds of integral

Apparently either I've forgotten some basic rule about integrals (it has been a while since I've taken a basic calc class) or something is wrong with this problem in pearson mylab. This was the ...
1
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2answers
68 views

Another integral (when cosine turns bad !)

I encountered a problem which was like this :- let $ I (m,n) $ be the indefinite integral of $ \int cos^{m}(x) cos(nx) dx $ . Then find $ I(5,7) $ in terms of $ I(4,6) $ and some function of sine ...
-2
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3answers
63 views

Why does $\lim\limits_{n \rightarrow \infty} 2 \int_{0}^{n} \frac {\tan^{-1} (x)} {n}\ dx = \pi$? [on hold]

Show that $$\lim\limits_{n \rightarrow \infty} 2 \int_{0}^{n} \frac {\tan^{-1} (x)} {n}\ dx = \pi.$$
9
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1answer
134 views

How to compute this improper integral?

Let $n\geq1$ be an integer and let $$I_n=\int\limits_{0}^{\infty}\dfrac{\arctan x}{(1+x^2)^n} \,\mathrm dx$$ Prove that $$\sum\limits_{n=1}^{\infty}\dfrac{I_n}{n}=\dfrac{\pi^2}{6} \...
1
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0answers
29 views

A double integral with a closed form, generalization

I have encountered a double integral with three parameters which has the following form: $$I(a,b,c)= \int_{-\infty}^\infty \int_{-c}^\infty \frac{e^{-a (x^2+b x+y)} \mathrm d y \mathrm d x}{\sqrt{(x^...
0
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0answers
26 views

Surface area of part of sphere $x^2+y^2+z^2=a^2$ enclosed by cylinder $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Note the given cylinder $a>b>0$ is elliptical. What I did: I took one fourth of the ellipse in the $xy$-plane and called it $K$, with $$K= \left\{ (x,y):~ 0 \le x \le a,~ 0 \le y \le b\sqrt{1-\...
0
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3answers
47 views

Prove that $\int_a^c f(x)\ dx = \int_a^b f(x)\ dx +\int_b^c f(x)\ dx$

Suppose $a,b,c$ are constants, $a < b < c$. The interval $[a,c]$ is in the domain of $f$. Prove that $$\int_a^c f(x)\ dx = \int_a^b f(x)\ dx+\int_b^c f(x)\ dx$$ This statement(property) is ...
0
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0answers
52 views

Evaluating $\int_1^A x^{-\alpha} dx$ as $A\to\infty$

Suppose $\alpha$ is a constant, $A$ is a positive real number. (a) Find the value of $\int_1^A x^{-\alpha} dx$, for $A > 1$. (b) What is the answer of (a) when $A$ tends to infinity? ...
0
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1answer
53 views

How can we prove that this integral converges?

If we have integral in the form $\int_0^\infty \frac{1}{(r+a \exp (-xt) )]\sqrt{(1+(a \exp (-xt))^2)}} \ dt $ and if we take difference of such two integrals with the same $r>0$ and different (or ...
0
votes
2answers
48 views

How to prove that $\lim\limits_{\epsilon\rightarrow 0}\int_{-\infty}^{\infty}f_\epsilon(x)g(x)dx=g(0)$ (Dirac delta function))

I'm currently studying the Dirac delta function using a textbook which unfortunately provides only partial solutions to its explanations. Why does $\lim\limits_{\epsilon\rightarrow 0}\int_{-\infty}^{\...
1
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1answer
31 views

Value of a definite integral changes when rearranging the integrand.

Consider $I=\int^{2\pi}_0\sqrt{1-\sin\phi}~\mathrm d\phi.$ A quick calculator check shows that the value of this integral is about $5.7$. If we now try to get the integrand into a form that can be ...
0
votes
0answers
22 views

Uncoditional formula for unsigned area under a linesegment w.r.t. $y=0$

I'm interested in finding the unsigned area under a line segment with respect to $y=0$. The line segment is defined by start point $(s_x, s_y)$ and end point $(e_x, e_y)$ Without loss of generality, ...
-2
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0answers
29 views

How to find primitive function of [on hold]

$$\int\frac{x\sqrt{x+1}}{x-\sqrt{x-1}}$$
0
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0answers
26 views

How to integrate over arbitrary quadrilateral

I need to integrate the product of two polynomial functions defined on an arbitrary (convex) planar quadrilateral defined by 4 points in $\mathbb{R}^3$. I was trying to firstly rotate the system of ...
0
votes
1answer
46 views

Vanishing of the integration along vertical line

2- Show that if $C$ is a vertical line segment $c \leq y \leq d,$ and if $F$ is a function of 2 variables defined on $C$, then $$\int_{C} F(x,y)dx = 0$$. I understand that the integration is ...
1
vote
1answer
36 views

For $g(x) = \int_0^\infty \frac{1}{x+y} f(y) \, dy$, Show $m\{ x \in (0,\infty) : g(x) > \lambda \} \le 1/\lambda \cdot \lVert f \rVert_{L^1}$

Q: For $x \in (0, \infty)$ let: \begin{align*} g(x) &= \int_0^\infty \frac{1}{x+y} f(y) \, dy \\ \end{align*} Show that for $f \in L^1(0,\infty)$: \begin{align*} m\{ x \in (0,\infty) : g(...
0
votes
2answers
21 views

if a particle moves at time t $-\pi<t<\pi$ which is given by x(t)=sin(3t) and y(t)=2t, how do i find total distance traveled?

If a particle moves at time t $-\pi<t<\pi$ which is given by x(t)=sin(3t) and y(t)=2t, how do i find the total distance traveled? While I can find the displacement using pythag theorem, I need ...
0
votes
0answers
10 views

Better method to find mass and center of mass of $S$ which is limited by the paraboloid $z=4x^2+4y^2$ and the plane $z=a>0$

So I need to find mass and center of mass of the solid $S$ limited by $z=4x^2+4y^2$ and the plane $z=a>0$. The solid is homogeneous with density = 1. I'm using cylindrical coordinates, to find ...
0
votes
0answers
32 views

How to solve this contour integral?

I was reading this where I encountered the following contour integral as given in equation (2.4) of the same. $$S = -i\int_{-\infty}^{+\infty} d\omega \log(\omega^2 + m^2 + E)$$ where $m,E \in \...
0
votes
1answer
38 views

Compare values of the three definite integrals given $f(x)=(\tan x)^\frac32-3\tan x+\sqrt{\tan x}$

Let $f(x)=(\tan x)^\frac32-3\tan x+\sqrt{\tan x}$. Consider the integrals $$I_1=\int_0^1f(x)dx$$ $$I_2=\int_{0.3}^{1.3}f(x)dx$$ $$I_3=\int_{0.5}^{1.5}f(x)dx$$ Then, prove that $I_1>I_3>I_2$...