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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1
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1answer
19 views

Periodic Function with Integral

Problem: $f(x)$ is a continuous function, and it is periodic with period $T$. For any $a<b$, prove that $$\lim_{n\to\infty}\int_a^bf(nx)dx=\frac{b-a}{T}\int_0^Tf(x)dx$$ I tried substituting ...
1
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0answers
24 views

Volume of a geodesic ball in a Riemannian manifold with $K<0$.

Let M be a simple connected Riemannian Manifold wih $K_M < 0$. Prove that the volume of any geodesic ball of M is strictly greater than $\frac{Vol(S^{n-1})r^n}{n}$, where $n = dim(M)$ and $r$ is ...
0
votes
1answer
57 views

More complicated uSubstitution

I have absolutely no idea what to do here other than use uSubstitution. $$\int{{1}\over{4x^2 + 9}}\mathrm dx$$ I also tried looking at the output of an integral calculator but to no avail. I noticed ...
2
votes
1answer
46 views

Need help with an integral involving the Dirac delta function

I'm having trouble evaluating this integral, which involves the Dirac delta function: $$ \int\limits_{0}^\infty \frac{\cos(\pi x)}{x} \delta \left[ (x^2-1)(x-2) \right] \mathrm{d}x $$ I think I ...
3
votes
3answers
116 views

Does $\int_0^{1/2} \frac{1}{x\ln x}dx$ converge?

I tried this: $$ \begin{align*} \ln x &= t \\ \frac{1}{x} dx &= dt \\ \lim_{x \to 0^+} \ln x &= -\infty \end{align*} $$ So now we have $$ \int_{-\infty}^{\ln(1/2)} \frac1t dt $$ which ...
0
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0answers
22 views

line integral problem, stuck in the curve

Problem: A force field is given in polar coordinates by the equation $$F(r,\theta)= (-4\sin (\theta), 4\sin (\theta)).$$ Compute the work done in moving a particle from the point $(1,0)$ to the ...
0
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1answer
23 views

Upper sum is strictly subadditive

Can you give me an example demonstrating that the upper sum is strictly subadditive (i.e. (the upper sum of $f$) + (the upper sum of $g$) is strictly bigger than the upper sum of $(f+g)$)?
4
votes
1answer
67 views

Proving that a function is integrable

Given that $f:[0, \infty] \to \mathbb{R}$ is decreasing with $\displaystyle\lim_{x \rightarrow \infty} f(x)=0$, prove that $$I=\int_{0}^{1}\frac{\cos(\frac{1}{x})f(\frac{1}{x})}{x^2}dx$$ converges. ...
1
vote
1answer
105 views

How to evaluate $\int\left\{\log\left(\log\left(x\right)\right) + {1 \over \left[\log\left(x\right)\right]^{2}}\right\}\,\mathrm{d}x$?

$$ \mbox{How to evaluate ?}\quad \int\left\{\log\left(\log\left(x\right)\right) + {1 \over \left[\log\left(x\right)\right]^{2}}\right\}\,\mathrm{d}x $$ Hints or suggestions please.
0
votes
0answers
56 views

Evaluating an integral

I have problem to evaluate the following integral: $$I=\int_{0}^{2\pi} \frac{\sin^2\phi\, d\phi}{a+b \cos{\phi}}$$ In fact I calculate it using complex integration methods, by changing the variable : $...
-1
votes
0answers
6 views

$e^{-x^2}M_{k,m}(x^2) \in L_1$ space?

Let $M_{k,m}(z)$ be m-Whittaker Function defined in enter link description here. How can show that $e^{-x^2}M_{k,m}(x^2),\, m,k > 0$ is belongs to $L_1\left(\mathbb{R}\right)$ or not? Thank you ...
-1
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2answers
42 views

Showing a function is integrable [on hold]

Let $\xi, \zeta\in\mathbb{R}^m$. How might one try to show that $f:\mathbb{R}^m\times\mathbb{R}^m\rightarrow \mathbb{R}$, defined by $\displaystyle\frac{1}{(1+\left|\xi - \zeta\right|)^{k}}$ is or is ...
14
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2answers
255 views

Daunting series of integrals: $\sum_{n=2}^\infty\int_0^{\pi/2}\sqrt{\frac{(1-\sin x)^{n-2}}{(1+\sin x)^{n+2}}}\log(\frac{1-\sin x}{1+\sin x})dx$

My coleague showed me the following integral yesterday \begin{equation} I=\sum_{n=2}^{\infty}\int_0^{\pi/2}\sqrt{\frac{(1-\sin x)^{n-2}}{(1+\sin x)^{n+2}}}\log\left(\!\frac{1-\sin x}{1+\sin x}\!\...
0
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3answers
84 views

Differential Equations (Coffee)

This is a long post so bear with me until I get to the part where I am stuck on! :) Question: The author of a popular detective novel drinks black coffee to help him stay awake while writing. ...
0
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1answer
21 views

Integral Inequality with Monotonic Function

Problem: For continuous, either both increasing or both decreasing functions $f, g$ on $[a, b]$, suppose that $p(x)$ is continuous and positive. Prove that $$\int_a^bp(x)f(x)dx \int_a^bp(x)g(x)...
3
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0answers
18 views

Defining derivatives and integrals for hyperoperations > 2

Derivatives and Integrals are continuous generalizations of the Forward Difference and Summation additive operators respectively. We can do the same with multiplication and get multiplicative calculus ...
0
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0answers
26 views

Fubini's theorem (interchange of sum and integrals) in case of multivariable function

Can the Fubini's theorem in case of single variable sequence of functions be readily extended to multivariable sequence of functions?, i.e, Is it true to say $$\iiint_V\sum_{n=0}^\infty f_n(u,v,w) \,...
1
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0answers
25 views

Kline Calculus intuitive approach Chapter 3 problem 12

The problem is as follows : Water drops flow out from a small opening at the rate of one drop per second and fall vertically with an acceleration of 32 ft/sec^2. Determine the distance between two ...
1
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1answer
33 views

Evaluate the integral $\int_{-\infty}^{\infty}{\left| 2t\cdot\text{sinc}^2(2t)\right|^2}\,dt$

I have a question in solving the integral $$\int_{-\infty}^{\infty}{\left| 2t\cdot\text{sinc}^2(2t)\right|^2}\,dt.$$ I know that you can use Parseval's Theorem to prove that $\int_{-\infty}^{\infty}\...
5
votes
3answers
266 views

Integration by parts or substitution?

$$\int_{}^{}x e^x \mathrm dx$$ One of my friends said substitution , but I can't seem to get it to work. Otherwise I also tried integration by parts but I'm not getting the same answer as wolfram. ...
0
votes
1answer
56 views

Compute definite integral by hand [closed]

How can I compute $$\int_0^1 \frac{x^3t}{(x^2+t^2)^2} \, \mathrm{dt}$$ by hand?
3
votes
2answers
47 views

Improper integral - checking convergence of $\int_{1}^{\infty} x^2 \sin(x^4) dx$

Does the following improper integral converges ? $$\int_{1}^{\infty} x^2 \sin(x^4) dx$$ Tried to find some known improper integral to compare this one to, but didn't find one. Thanks for helping!
10
votes
3answers
231 views

How to prove that$\int_{0}^{1}\ln{(x/(1-x))}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$

$$\int_{0}^{1}\ln{\big(\frac{x}{1-x}\big)}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$$ Put $$\frac{x}{1-x}=y$$ $$I=\int_{0}^{\infty}\ln{y}\ln{(1+3y+y^2)}\frac{dy}{y(y+1)}=\frac{8}{5}\zeta{(3)...
1
vote
2answers
52 views

Given $f(x,y)$ is a continuous function, Do these integrals equal? [closed]

Given range $\{ 0 \le x \le 1, 0 \le y \le 1\}$ Do these integrals equal? $\int_0^1(\int_0^y f(x,y)dx)dy = \int_0^1(\int_0^x f(x,y)dy)dx$ Well, the answer is no. It seems like the triangulars are ...
2
votes
2answers
67 views

Closed form for an integral with log and power

Let $n \in \mathbb{N}$. We know that: $$\int_0^1 x^n \log(1-x) \, {\rm d}x = - \frac{\mathcal{H}_{n+1}}{n+1}$$ Now, let $m , n \in \mathbb{N}$. What can we say about the integral $$\int_0^1 x^n \...
0
votes
0answers
21 views

numerical integration asymptotic relation

Let $Q\subset R^n$ be a convex subset and $f\in C^2(Q)\;$ We set $x_s:=\int_Q xdx$,$\;\;\;Vol(Q):=\int_Q 1dx$ and $diam(Q)=sup||x-y||_2$ Prove the following asymptotic relationship: $...
6
votes
1answer
106 views

Evaluate $\int \frac {\sin(x)}{x^2 + 4x + 5}dx$

Question: Evaluate $$ \int \frac{\sin(x)}{x^2 + 4x + 5} dx=\int \frac {\sin(x)}{(x + 2)^2 + 1}dx $$ By using the change of variable $y = x + 2$ we have that $dy = dx$ then $$I = \int \frac{\...
3
votes
4answers
87 views

Showing that $\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$

Where n is an integer, $n\ge1$ and $(A,B)$ just constants $$I=\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$$ It is obvious that $$\int_{-n}^{n}x+\tan{x}dx=0$$ Let make a ...
-1
votes
5answers
88 views

Evaluation of $\int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx$

Evaluation of $\displaystyle \int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx$ $\bf{My\; Try::}$ Let $$I = \int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx\;,$$ Put $x+1=t\; $ Then $dx = dt$ and changing limits, we get ...
1
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0answers
23 views

Matlab double integration result does not match with my self calculate

Here is a double integration, self learning $$ P=\int_{-w}^{w}\int_{l}^{\frac{y_h(x_b+w)}{x_h}+l}\frac{1}{2}\operatorname{erfc}\left[\frac{\log{\frac{z_h(y_b-l)}{y_h}}-\mu}{\sigma\sqrt2}\right]\space ...
-2
votes
1answer
41 views

What are the solutions to integration problems below? [closed]

$$\int (a^2 - y^2)y dy $$ $$\int \frac{e^{\sqrt{x}} + 1}{\sqrt{x}} \ dx$$ $$\int \frac{x^3}{\sqrt{1-2x^2}} \ dx $$
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votes
0answers
34 views

show the if f(x)'>0 the integral exist, help please. [closed]

I need to show that if f(x) is Superior monochrome in [a,b] the integral for f(x) in [a,b] exist, from here I know that if f(x) is Superior monochrome for all x2,x1∈[a,b], (x2>x1), f(x2) ≥ f(x1) I ...
1
vote
1answer
41 views

Parallelizable open dense subset and integration

In Petersen's Riemannian Geometry (2016), it is stated on page 8 that any manifold $M^n$ has an open dense subset $O$ with $TO=O\times\Bbb R^n$. Thus it is orientable and one may define the integral ...
0
votes
3answers
59 views

How to find a parametrization for a torus?

I need to compute the surface area of the torus $$T^2=\{(x,y,z)\subseteq\mathbb R^3 \left(\sqrt {x^2+y^2}- R\right)^2+z^2=r^2\}$$ where $0<r<R$. I know I need to compute the metric tensor and ...
2
votes
1answer
27 views

Integrating a composition

I'm need to calculate this: $$\int g'(x) (2g(x) - \frac{1}{g^2(x)}) dx $$ I think i have to integrate by parts, so i put: $$ dv= g'(x) dx,v=g(x)$$ $$u=2g(x) - g(x)^{-2},du=2g'(x)-(g(x)^{-2})'$$ ...
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votes
0answers
41 views

Prove for integrals the following [duplicate]

If n is a positive integer, prove that $\int_0^n{[t]^2 dt} =\frac{n(n-1)(2n-1)}{6}$. Please prove it in a descriptive way.
1
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1answer
42 views

Produce a sequence $(g_n):g_n(x)\ge 0$ and $\lim g_n(x)\neq 0$ but $\int_{0}^{1} g_n\to 0$

Produce a sequence $(g_n):g_n(x)\ge 0,\,\forall x\in [0,1],\,\forall n\in\Bbb N$ and $\lim g_n(x)\neq 0,\,\forall x\in [0,1]$ but $\int_{0}^{1} g_n\to 0$ Im in need to clarify that Im talking of ...
0
votes
0answers
34 views

How to calculate the line integral $\int_{\vec{\gamma_j}}\langle\vec{v_k},d\vec{x}\rangle$ for $j=1,2$ and $k=1,2,3$

Here's the integral again: $$\int_{\vec{\gamma_j}}\langle\vec{v_k},d\vec{x}\rangle$$ Here's what I know about $\vec{\gamma_1},\vec{\gamma_2}:[0,1]\rightarrow\mathbb{R^3}$ $$\vec{\gamma_1}(t) = \left(...
5
votes
1answer
101 views

finding the series $\sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$

My goal is to solve this series $$S(x) = \sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$$ I did took the derivative first w.r.t $x$ $$S'(x) = \sum_{n=1}^\infty \frac{x^{n-1}}{n!}$$ which I ...
0
votes
3answers
59 views

Proving a Definite Integral Inequality without Geometrical Intuition

I solved an integral inequality problem using geometrical methods. However, I just cannot satisfy with them and want a without-geometrical-intuition proof, and I couldn't find one. Proof the ...
0
votes
2answers
60 views

How to substitute in an integral

I need to solve this integral by substitution: $$ \int \frac{1}{(1+\sqrt x)^2}\,dx $$ I know the substitution should be: $$ u= \sqrt x$$ and so $$du=\frac{1}{2\sqrt x}$$ but i can't understand how ...
0
votes
1answer
45 views

Calculation of double integral

I am trying to solve this integral $$ \int_{20}^{21}\int_{20}^{25}\frac{1}{\sqrt{2\pi}ga_{m}}\exp\Big{(}-\frac{1}{2} \frac{(a_{m}-(ba_{f}+c))^{2}}{g^{2}a_m^{2}}\Big{)}da_{m}da_{f} $$ with b,c,g ...
1
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0answers
46 views

Simplify an integral or the integrand involving hyperbolic functions

I would like to simplify the following integral or at least the integrand: $$f(t):=\int_{a}^{t-a} \frac{(\cosh(t-x)-\cosh(a))^{i\tau-1/2}}{(\cosh(x)-\cosh(a) )^{i\tau+1/2}} dx, \quad t>2a,$$ ...
-1
votes
1answer
57 views

How to implement twice MATLAB integral build-in function for numeric integration? [closed]

Suppose we have a function $F(\lambda) = \int\limits_{\lambda}^1 f(x) dx$, where $f(x)$ has no formula for antiderivative. We can easily calculate it by means of build-in MATLAB functions. Let's use $...
1
vote
0answers
11 views

Mean continuity of gradient

Let $f:\mathbb R^n\longrightarrow R$ be a differentiable function, and suppose $\nabla f$ is bounded. Prove that $$\lim_{r\to 0}\frac{1}{\omega_n r^n}\int_{B_r(x)}[\nabla f(y)-\nabla f(x)] dy=0.$$ ...
0
votes
1answer
54 views

How does one compute this heavy integral?

The integral is $$\frac{1}{2\pi i}\int_\Gamma\frac{\exp(z^2-\cos(iz)-4)}{z-2}dz$$ where $\Gamma$ is the unit circle. Here's how I tried to parametrize it: $z=e^{i\theta}$ on $\theta\in [0, 2\pi]$, ...
3
votes
3answers
84 views

Volumes of Revolutions : Lord of the Rings

Question: The "Lord of the Rings" has a collection of solid gold rings for different-sizes fingers. The cross section of each ring is a segment of a circle radius $R$ as shown in the diagram below. ...
0
votes
1answer
43 views

Integration of Rational function contain even power of variable

Evaluation of $\displaystyle \int\frac{1}{1+x^6}dx$ $\bf{My\; Try::}$Let $$I = \int\frac{1}{1+x^6}dx = \int\frac{1}{(1+x^2)(x^4-x^2+1)}dx$$ Using Partial fraction , above Integral is very lengthy, ...
0
votes
5answers
51 views

Integration with limits and options.

I found this exercise in an old exam but I don't know how to attack it because is a limit of an integration and I don't know if the limit affects the process of the integral or it makes it easier. The ...
0
votes
0answers
14 views

Multidimensional integration help

Please confirm my understanding of a function I would like ultimately to plot: $C_{ij}(l)=\int^{\chi_{h}}_{0}W_{i}(\chi)W_{j}(\chi)\frac{P(k=\frac{l}{\chi};z)}{\chi^{2}}d\chi$ where the weights with ...