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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0answers
140 views

Implicit function theorem to prove tangent plane to the surface

Let $\Phi$ be the regular surface at $(u_o,v_o)$ (ie., $\Phi$ is of class $C^1$ and $T_u\times T_v\ne 0$ a)Use the implicit function theorem to show that the image of $\Phi$ near $(u_o, v_o)$ is the ...
7
votes
2answers
179 views

How to find closed-form of $\int_{0}^{+\infty} \operatorname{sech}^2 (x^2)\,dx$

How to find this integral closed form: $$I=\int_{0}^{+\infty}\operatorname{sech}^2{(x^2)}\,dx$$ where $\operatorname{sech}{(x)}$ is defined as secant of hyperbolic function. This problem ...
0
votes
1answer
27 views

An Integral Inequality Question

We have the functions $f$ and $g$ such that, $$f:\mathbb{[0,1]}\rightarrow\mathbb{R}$$ $$g:\mathbb{[0,1]}\rightarrow\mathbb{R}$$ and both $f$ and $g$ are bounded and continuos ...
2
votes
0answers
45 views

The best student

Suppose two students called A and B. Student A has answered $k_A$ questions correctly out of $n_A$ questions. Student B has answered $k_B$ correctly out of $n_B$ questions. Who is the best student ...
4
votes
1answer
90 views

Find this sum $S$ using Real-analysis methods only

$$S = \sum_{k=1}^{\infty}\frac{2H_k}{(k+1)(k+2)^3}$$ I have tried a lot and failed, any help is appreciated. $H_k$ is the harmonic number. Thanks (real method only please)
1
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0answers
39 views

Multiplying two sums?

(Real-analysis only) I will admit, I have posted a question similar to this, but this question's aim is to ask how to multiply the sum and integrate it. $\displaystyle \log^2(x) = ...
0
votes
1answer
51 views

Mathematical problem with solving a physics issue

The power absorbed by the BOX in the Fig.A is $p(t) = 2.5e^\left(-4t\right)$ W. Compute the energy and charge delivered to the BOX in the time interval $0 < t < 250$ ms. So let me show ...
0
votes
2answers
66 views

Can't do basic integral

I seem to be brain dead today, and I can't do the following integral: $\displaystyle\int \cfrac{1}{\sqrt{-cx^3}} dx$ Why is this wrong: $$\displaystyle\int \cfrac{1}{\sqrt{-cx^3}} dx= ...
0
votes
2answers
53 views

Does L1 and nonnegative imply bounded almost everywhere?

Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ a nonnegative function, such that $f\in L^1(\mathbb{R})$. Does this imply that $f$ is bounded almost everywhere?
6
votes
2answers
175 views

Closed form for a zeta series

It is not that diffcult to derive \begin{align} \sum^\infty_{k=2}\frac{(-1)^{k-1}\zeta(k)}{k2^k}=&-\frac{\gamma}{2}+\ln\left(\frac{2}{\sqrt{\pi}}\right)\tag{1}\\ ...
0
votes
1answer
43 views

Application of the fundamental theorem of calculus. $F(x)=\int_a^x f(t)(x-t) dt$ evaluate $F''(x)$

Let $f$ be a continuous function on $\Bbb R$ $F(x)=\int_a^x f(t)(x-t) dt$ Evaluate $F''(x)$ I used the fundamental theorem of calculus to attempt this question. My attempt at the question ...
2
votes
0answers
68 views

Evaluate $\int_{0}^{1} \log^2(x)\log^2(1-x)\,dx$ using non-elementary methods. [duplicate]

The task is to evaluate, using the specific given hint, and only real-analysis methods, no complex analysis is allowed in the problem. $$\int_0^1 \log^2(x)\log^2(1-x)\, dx$$ The hint given is: $$ ...
1
vote
2answers
112 views

Volume inside 3 cylinders

Find the volume of the region lying inside all three of the circular cylinders $$x^2+y^2=a^2,$$ $$x^2+z^2=a^2,$$ $$y^2+z^2=a^2$$ Hint: Make a good sketch of the first octant part of the region, and ...
0
votes
1answer
23 views

Obtaining the integral:$\int_{0}^{\infty} e^{-st}f^n(t)dt$

I have the following integral in my notes: $$\int_{0}^{\infty} e^{-st}f^n(t)dt=\Big[ ...
2
votes
0answers
54 views

Can the minimum be given by an integral?

for $a,b > 0$, $$ \begin{align} &\int_{0}^{\infty} \frac{\sin (ax) \sin (bx)}{x^{2}} \ dx \\ &= \int_{0}^{\infty} \frac{a \cos (ax) \sin (bx) + b \sin(ax) \cos(bx)}{x} \ dx \\ &= ...
1
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0answers
45 views

A question related to Integral and supremum

Let $f\in L_{p}([0,1])$ and 1-periodic on $R^{1}.$ Suppose $[a,c]\subset [0,1].$ Are the following quantities equal? $$ \underset{|h|\leq \delta_{1}}{\sup}\int_{a}^{b}|f(x+h)-f(x)|^{p}dx+ ...
1
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0answers
39 views

Moments of $|ax-by|$

Suppose that $X$ and $Y$ are independentr.v. uniform on $[0,1]$. What is the $E[|aX-bY|^p]$ for some constants $a,b,p>0$? What I did. \begin{align*} E[|aX-bY|^p]=\int_0^1\int_0^1|ax-by|^p dx ...
0
votes
2answers
60 views

A change of variable to express as Lower incomplete Gamma function

THe lower incomplete gamma function is written as $$\Gamma(a,x)=\int_{x}^{+\infty} t^{a-1} e^{-t}dt $$ I have found read in paper that the following integral $$\int_{x}^{\infty}e^{-t^p}dt ...
0
votes
2answers
62 views

What is the integral of $\int_0^1 \int_0^1 |x-y| dx dy$

What is the integral of $\int_0^1 \int_0^1 |ax-by| dx dy $ for some constants $a,b$. I was thinking to do split in on when it's positive and when it's negative like this \begin{align} \int_0^1 ...
1
vote
1answer
52 views

Integral by polar coordinates

How to calculate the integral $$\int_0^6\int_0^y x\;dx dy$$ using polar coordinates?$$$$I know that $x=R\cos \theta$ and $y=R\sin\theta$ and that the Jacobian is $R$.
1
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3answers
68 views

Indefinite integral of $\frac{\arctan x}{x^2+1}$

EDIT: I was studying from a site that uses really ambiguous notation so I misread $\arctan\ (x)^2$ as $\arctan\ (x^2)$. Now I can see why the integral is actually $\frac{1}{2} \arctan^2\ x + c $. ...
4
votes
2answers
172 views

Computing $\sum_{n=1}^{\infty} \frac{\psi\left(\frac{n+1}{2}\right)}{ \binom{2n}{n}}$

Here is an interesting series I played with, namely $$\sum_{n=1}^{\infty} \frac{\displaystyle\psi\left(\frac{n+1}{2}\right)}{\displaystyle \binom{2n}{n}} \approx -0.245969181104090562617616399148$$ ...
1
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0answers
67 views

Evaluation of $\int_{0}^{\pi} \lfloor \cot x\rfloor dx$.

Evaluation of $\displaystyle \int_{0}^{\pi} \lfloor \cot x\rfloor dx\;,$ Where $\lfloor x \rfloor $ is floor function of $x$. $\bf{My\; Solution ::}$ Let $\displaystyle I=\int_{0}^{\pi} \lfloor ...
4
votes
1answer
73 views

Understand this Fourier transform $\int \frac{1}{|x|}e^{ikx} d^3 x = \frac{4 \pi}{k^2}$

I found the equation $$\int \frac{1}{|x|}e^{ikx} d^3 x = \frac{4 \pi}{k^2}$$ in a 'physics' textbook and I just don't understand what this equation tries to tell me. Is there anybody who ...
1
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3answers
75 views

Solve $\frac{dr}{d \theta}+r\tan \theta =\frac{1}{\cos \theta}$

We are given the differential equation $\frac{dr}{d \theta}+r\tan \theta =\frac{1}{\cos \theta}$ And we are asked to find a solution. I'm having difficulties isolating $r$ and $\theta$ to different ...
8
votes
1answer
254 views

Evaluate $\int_{-1}^{1} \exp(x+e^{x})\,dx$

Evaluate $$\int_{-1}^{1} \exp({x+e^{x}})\,dx$$ where $\exp(x)=e^x$. Can anyone give me any tips on where to start with this? I've tried doing it be substitution, with $ u=e^x$ and ended up needing ...
0
votes
1answer
39 views

Is this a solution to the ODE - simple ODE question

We are given the first order linear differential equation: $y'-2xy=1$ We have guessed a solution to the ODE: $y=e^{x^2}\int e^{-t^2}dt+e^{x^2}$ And we are asked, is this a valid solution to the ode ...
1
vote
1answer
63 views

Riemann integrable function; approximation by step functions

From the definition of a Riemann integrable function $f : [a,b] \to \mathbb R$ it follows that there exist sequences of step functions $\varphi_k$ and $\varphi^k$ with $\varphi_k \leq f \leq ...
1
vote
2answers
47 views

When would you want to model a derivative?

I just read this, and am intrigued. http://formulize.nutonian.com/documentation/eureqa/tutorials/modeling-derivatives/ What kind of model would you have a scatter plot of data points, and want to ...
0
votes
3answers
77 views

Evaluate $\int\cos^3x\sin^2x\,dx$

My university mathematics is kind of poor. But as I am learning advanced mathematics this becomes a major shortcoming. I tried to integrate $\int$$\sin^3\theta$$\cos^2\theta$d$\theta$ = ...
0
votes
2answers
144 views

How to evaluate $\int _{-\infty }^{\infty }\!{\frac {\cos \left( x \right) }{{x}^{4}+1}}{dx}$

How to evaluate the following integral? $$ \int _{-\infty }^{\infty }\!{\frac {\cos \left( x \right) }{{x}^{4}+1}}{dx} $$ Unlike this example, according to maple, the solution does not contain sine ...
1
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2answers
60 views

Integrating of product of sine and cosine

$$ \int_L^{-L} \cos{\frac {n \pi x}{L}} \sin{\frac {m \pi x}{L}} dx = 0 $$ for any $n$ and $m$ . I did integrating by parts, but I am not getting this equals to $0$. Can anyone show this?
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0answers
55 views

integrate over quadrant of sphere

It is known that $\int_{x\in S}\exp(\kappa\mu^Tx)dx$ where S is the surface of the unit sphere is $\frac{(2\pi)^{p/2} I_{p/2-1}(\kappa) }{\kappa^{p/2-1}}$ where $p$ is the number of dimensions and $I$ ...
0
votes
1answer
33 views

What is the joint CDF of f(x,y)=2(x+y) 0<=x<=y<=1

I am trying to find the joint CDF of $f(x,y)=2(x+y) : 0\leq x\leq y\leq 1$. There are five different answers for the CDF depending on the restrictions of $x$ and $y$ that you use. I found the CDF ...
0
votes
0answers
152 views

Inclined Elliptic Tank Volume Calculation

Can someone help me determine an equation for calculating the volume of an elliptical cylinder on it's side and inclined 5 degrees from horizontal? The tank has flat ends. I have found several ...
0
votes
1answer
41 views

What algorithm to solve this integral similar to a normal CDF numerically?

I'm looking to solve this integral numerically. It is a bit similar to a normal CDF. z and tau are deterministic. What kind of algorithm may do the job, ideally to be coded in C/C++? I'm ...
1
vote
1answer
44 views

Finding the antiderivative of a real power of a rational function

my abilites in integration (applied) are very limited, so my question is: if i would like to find out how to approach a problem like finding the antiderivative of the function $f(x) = ...
0
votes
1answer
28 views

How to take the integral of a derivative to obtain desired result?

I am aiming for the form of derivative below computed over time that causes its differentiated variable V to go from an initial -.001 and increase to reach 10. I will explain my current calcs below ...
3
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1answer
85 views

Integrating tensors on manifolds

When/how can you integrate tensors on manifolds and what does it mean? I imagine that line integrals of tensors make sense when you have a connection, since you can uniquely parallel transport all ...
0
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4answers
84 views

$\int \sqrt{x} 2^{-\sqrt{x}}dx$. How to begin?

Let's take: $$\int \sqrt{x} 2^{-\sqrt{x}}dx$$ I don't know how to begin, I am asking for advice.
2
votes
2answers
63 views

How to determine this integral [duplicate]

Let's take: $$\int \frac{1}{x\sqrt{x+1}} \ dx $$ I tried solve this by four hour, so I am asking for help
1
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0answers
30 views

Contour Integration Problem for reciprocal of Bessel Function Power

Hopefully some one can help me with this problem. I have to compute $$ \oint_C \frac{1}{[J_0(\alpha t)]^n}dt$$ where $C$ is a contour that contains the first zero ...
1
vote
2answers
73 views

What is the difference between $\delta x$ and $dx$

Sometimes I see the $\delta x$ and $dx$ but I don't know exactly what is the difference between them.
6
votes
1answer
114 views

Find the closed form of the digamma related series

The question I asked here Computing $\sum_{n=1}^{\infty} \left(\psi^{(0)}\left(\frac{1+n}{2}\right)-\psi^{(0)}\left(\frac{n}{2}\right)-\frac{1}{n}\right)$ made me think to ask for your support for ...
4
votes
1answer
133 views

Contour integration of $\int_{-\infty}^{\infty}\frac {\sin^3 x}{x^3} dx$: where are the singularities?

I have just begun to study complex analysis and I'm trying to calculate $$ \int_{- \infty}^{\infty} \frac {\sin^3 x}{x^3} dx $$ with the "help" of an exercisebook. I have followed these steps: ...
0
votes
0answers
36 views

Calculus question table leg

Calculate the volume of the “table leg” for which the radius r at height z is given by the following expression, with $0 \leq z \leq 12 $: $$r(z)=4+\cos \left( \frac{\pi z}{2} \right)$$ then it ...
0
votes
3answers
67 views

Evaluating $\int_{0}^{1} \frac{1}{(1+x^2)^{5/2}}dx $

$$\int_{0}^{1} \frac{1}{(1+x^2)^{5/2}}dx $$ it says to let $x = \tan(u)$ which i presume wants $$\frac{dx}{du} = \sec^2(u) = $$ $$dx = \sec^2(u) du $$ $$\int_{0}^{1} ...
0
votes
1answer
40 views

Evaluate $\displaystyle\iiint_W x^2 \cos z\,dv$

$$\displaystyle\iiint_{W}{x^2\cos z \ dv}$$ Where $W$ is the region bounded by $z=0, z=\pi, y=0, y=1,$ and $x+y=1$. I drew the region $W$ at home and found that it is a uniform triangular prism of ...
0
votes
1answer
46 views

Relation between sum and integral

I have an exercise (from physics) where I am supposed to show $$\sum_{k'<k_f} \frac{1}{|k-k'|^2} = C \left( \frac{1}{2} + \frac{1-(\frac{k}{k_f})^2}{4 \left( \frac{k}{k_f} \right) } ln |\frac{1 + ...
1
vote
2answers
49 views

Integral Measures: Variation

Given a measure $\lambda\geq0$. Regard a real function $h:\Omega\to\mathbb{R}$ with $h\in\mathcal{L}$. Consider the real measure $\mu(E):=\int_E h\mathrm{d}\lambda$. Then its total variation ...