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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-4
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0answers
30 views

How to calculate the following integral? Can you show me step by step!

How to calculate the following integral: $$\int_c^d \sqrt{a^2\sin^2x+b^2\cos^2x} dx$$?
0
votes
2answers
7 views

Given the integral of an equation over one set of bounds find the integral over another set of bounds.

If $\int_{1}^{3}f(w)dw=7$, find the value of $\int_{1}^{2}f(5-2x)dx=7$ I think this problem has something to do with the fact that (5-2(2)) = 1 and (5-2(1)) = 3 and these are the bound of the ...
5
votes
2answers
48 views

Is $\int_1^{\infty}\frac{x \cos(x)^2}{1+x^3}$ convergent or divergent?

For the integral $$I= \int_1^{\infty}\frac{x \cos^2(x)}{1+x^3},$$ how do I test this for convergence or divergence? I know that this an improper integral- however it cannot be solved so would need to ...
3
votes
1answer
35 views

Question about steps/derivation regarding Laplace method.

I am reading something on the Laplace method of integrals and I don't understand part of it's argument. It gives the integral $$\int_{-3}^4 e^{-\lambda x^2}\log(1+x^2)dx$$ and finding the leading ...
0
votes
0answers
11 views

expected length of a k-dimensional normal random vector

Let $X$ is a random vector size $p$ from multivariate normal distribution $\mathcal{N}$($0$, $\sigma$ $I$), $I$ is identity matrix. I want to find the expected value of reciprocal of norm like this ...
2
votes
1answer
45 views

Inverse Fourier transform of $\exp(4\pi^2i|\xi|^2t)$

I would like to compute the inverse Fourier transform of $\exp(4\pi^2i|\xi|^2t)$. \begin{equation} f(x,t) = \int_{\mathbb{R}^n} e^{2\pi i x\cdot\xi} e^{4\pi^2i|\xi|^2t} \,\mathrm{d}\xi \end{equation} ...
0
votes
0answers
19 views

division of two integrations

I am new to calculus. Today when I read the exponential family, The exponential family are defined as below: $$ p(x|\alpha) = h(x)exp\{ \alpha T(x) - A(\alpha)\}$$ $ T(x) $ is referred to as ...
2
votes
1answer
22 views

$ \lim_{t\to 0} \int_{|x|>\epsilon} \frac{e^{-\frac{x^2}{2t}}}{\sqrt{2 \pi t}} dx=0 $

I want to prove that: \begin{equation} \lim_{t\to 0} \int_{|x|>\epsilon} \frac{e^{-\frac{x^2}{2t}}}{\sqrt{2 \pi t}} dx=0, \end{equation} for any $\epsilon >0$ I've shown using polar ...
0
votes
1answer
40 views

Proving $\frac{\sin\pi z}{\pi z}=\prod_{n=1}^{\infty}\Big(1-\frac{z^{2}}{n^{2}}\Big)$

I apoligize if this has been answered already; the quick searches I've done have proven fruitless. I'm given that $\displaystyle\pi\cot(\pi z)=\frac{1}{z}+\sum_{1}^{\infty}\frac{2z}{z^{2}-n^{2}}$ ...
-1
votes
0answers
20 views

integral involving pdf of normal distribution

I want to calculate following integral any solution (closed form, or in terms of numerical functions, even approximations will help)
1
vote
0answers
39 views

Evaluate $\int_0^1 \int_\sqrt{y}^1 \int_0^{x^2+y^2} dz dx dy$.

Evaluate $\int_0^1 \int_\sqrt{y}^1 \int_0^{x^2+y^2} dz dx dy$. Attempt: $$ \int_0^1 \int_\sqrt{y}^1 \int_0^{x^2+y^2} dz dx dy = \int_0^1 \int_\sqrt{y}^1 x^2 + y^2 dx dy = 1/3 + 1/3 - 2/15 - 2/7 ...
0
votes
0answers
24 views

Extra Credit Triple Integral Inequality Proof

I'm working on the extra credit for my Calculus class and the last problem is a proof. We have done proofs before, but I'm unsure of how to approach this problem. Any help would be much appreciated, ...
0
votes
1answer
19 views

Torus coordinates and triple integration

Torus coordinates are given by $$x=(R+r \cos\theta)\cos\phi,\space y=(R+r \cos\theta)\sin\phi,\space z=r\sin\theta,$$ with $R>0$. A given torus $T_b$ is defined by $$T_b=\{(r,\theta,\phi):0\leq ...
2
votes
1answer
80 views

How do I solve this integral ? $\lfloor \rfloor$

I haven't come across this yet, but the question is to solve this definite integral: $$\int_{e}^{4+e} (3x- \lfloor 3x \rfloor)dx$$ What is obviously causing problems is the whole part of the number ...
-4
votes
1answer
27 views

Evaluate the integral by making the given substitution. [on hold]

Evaluate the integral by making the substitution $u=1-9t$. $$\int {1\over(1-9t)^3}\,dt$$
4
votes
2answers
72 views

Is there a probability distribution with mean $1$ such that $f(x)=\frac{1}{x}f\left(\frac{1}{x}\right)$

Is there a probability distribution defined over $\mathbb{R}^{+}$ by the pdf $f$ such that, $$\forall x > 0, f(x)=\frac{1}{x}f\left(\frac{1}{x}\right)$$ and $$\int_0^{\infty} x~\mathrm{d}f = 1 $$ ...
0
votes
0answers
9 views

Integration of the incomplete beta function

I would like to know if there is a way of computing the following integral analytically ($B_u$ is the incomplete beta function): $$\int B_u(a-1,0)~u^{-a} du$$ Thanks for your ideas.
2
votes
5answers
117 views

Finding $\int_{0}^{\infty }\frac{1}{1+x^4}dx$ [duplicate]

finding $$\int_{0}^{\infty }\frac{1}{1+x^4}dx$$ My attempt is: let $x=\sqrt{u}$ $dx=\frac{1}{2\sqrt{u}}$ $$\int_{0}^{\infty }\frac{1}{2\sqrt{u}(1+u^2)}du$$ here I stopped because I don't know how to ...
0
votes
2answers
40 views

Evaluating $\int_{-\infty}^\infty e^{tx^2} \frac{1}{\sqrt{2\pi} \sigma} \exp [ \frac{-x^2}{2\sigma^2} ] \ dx$

I have so far $$M(t) = E(e^{tX^2}) = \int_{-\infty}^\infty e^{tx^2} \frac{1}{\sqrt{2\pi} \sigma} \exp \left[ \frac{-x^2}{2\sigma^2} \right ] \ dx = \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}\sigma} ...
1
vote
1answer
8 views

forms of the Romberg Method equation

My teacher wrote the this equation for the Romberg method $ I_{j,k}=\frac{4^j I_{j-1/k+1}-I_{j-1/k}}{4^j-1} $ Is this the right equation? Most the equations I looked at online for the Romberg ...
0
votes
1answer
58 views

Evaluating $\int \frac{x^2}{1+e^{-x}}dx$

Consider $$\int \frac{x^2}{1+e^{-x}}dx$$ I've tried every method and trick that I'm familiar with, except by parts, but I can't seem to be able to acquire an elementary integral. Does there exist ...
3
votes
0answers
26 views

Differential Equation (Non linear to linear differential equation)

Show that the substitution $u=\frac{1}{y}$ transform the non-linear differential equation $$\frac{dy}{dx}+\frac{y}{x}=y^2\ln (x)$$ into the linear differential equation ...
1
vote
0answers
28 views

Complex Line integral of 1/z over the principle branch cut

I would appreciate it if someone checked my work to ensure that it's consistent. Compute the integral $\int_{C}\frac 1 z {dz}$ by obtaining an appropriate branch of the logarithm. There's an ...
1
vote
0answers
37 views

Integral Identity

A question from a multivariable calculus exam: I have tried lots of methods like integrating the RHS by parts. Any help would be appreciated. Find $w(y)$ such that the identity $$ ...
1
vote
1answer
23 views

Gravitational force inside a uniform solid ball - evaluation of the integral in spherical coordinates - mistake

I have been reading this PDF document: www.math.udel.edu/~lazebnik/BallPoint.pdf While trying Case A I found a small error (a $2$ was missing) but I was able to follow the argumentation and got to ...
2
votes
1answer
18 views

Evaluating a triple integral in spherical coordinates

I need to evaluate the integral $\int \int \int \frac{x^2}{x^2+y^2}$ over the region $D$ where $D = {(x,y)} : 1\leq x^2+y^2+z^2 \leq 2, z^2>=x^2+y^2$ and $z\leq 0$ So I tried converting to ...
0
votes
3answers
52 views

Does this integral converge or diverge?

I have the $$\int_{16}^{500} \frac{1}{x^{0.25} - 2} dx,$$ and am trying to find whether it converges or diverges. I have sketched the graph and noticed that their is an asymptote at $x=16$ (hence ...
1
vote
1answer
43 views

Weird integration

$$\int\frac{v}{(v+1)^2}dv=\int xdx$$ I saw this question in a math site. I don't have any idea to solve this. Can anyone guide me? Thanks in advance.
1
vote
1answer
37 views

Integral of periodic function.

I was wondering if anyone can help me with this, if f(x) is a periodic function with period T then it satisfies $$\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$$ for all $a \in \Bbb R$. It is clear that ...
1
vote
0answers
24 views

how to integrate this equation with two sums inside?

i'm reading a book, and i have trouble with this problem, i don't how to integrate this equation and where to begin from. Although they already give the answer but I don't understand how to get it. I ...
0
votes
0answers
30 views

Help with when the integral is convergent

quick question, how can you tell for which a the integral $$\int_1^{\pi/2} \dfrac{\cos^2(2x) - e^{-4x^2}}{x^a\tan{x}}dx$$ is convergent? Thanks in advance.
1
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0answers
23 views

Asymptotics of integrals

I am struggling with this problem where we're asked to use method of steepest descent: Find the leading term of the asymptotics of the following integral for $\lambda\to\infty$ : ...
3
votes
2answers
20 views

Showing that $\lim \int \left(\sum_1^n |f_k|\right)^p \le \left(\sum_1^\infty \|f_k\|_p\right)^p$

I am reviewing a proof about the completeness of $L^p$ spaces. The proof begins as such (Folland Theorem 6.6): For $1 \le p < \infty$, suppose $\{f_k\} \subset L^p$ and $\sum_1^\infty \|f_k\| = ...
1
vote
1answer
24 views

What is this Toeplitz like matrix called and how do I represent it as a convolution?

I have a matrix that is used to represent the Green's function in a popular class of fast E & M solvers (CG-FFT). The matrix represents distances, that are later filled in using the appropriate ...
1
vote
1answer
19 views

Mass and center of mass using double integrals

Disclaimer: This was given as a homework from college but the teacher didn't teach us anything about density or mass or anything related. A lamina has the form of the region limited by the parabola $ ...
-3
votes
0answers
17 views

analysis, rieman, integration, real analysis [on hold]

f is an integrable function in a compact set [a,b] -> R. Prove if integral((f(x)^2) over the set [a,b] is equal to zero then f(x0)=0 for any x0 in the compact. what you can say about the set {x in ...
0
votes
2answers
20 views

On the horizontal integration of the Lebesgue integral

I'm studying Lebesgue integral and its difference with respect to the Riemann one. I'm reading that the key difference (at least graphically speaking) is that the first slices the function ...
2
votes
2answers
35 views

Finding F ' (y)

This question looks deceivingly simple to me so I was wondering if someone out there could enlighten me whether it really is what I think it is or if I have completely missed the point. Let $$F(y) = ...
0
votes
0answers
17 views

Explain the formula of energy in signal processing [duplicate]

Please, give me intuitive understanding of this formula (http://en.wikipedia.org/wiki/Energy_%28signal_processing%29): So t is time, x(t) - signal function, integral is sum of this function on ...
1
vote
0answers
16 views

Heuristic: Daniell integral vs. Lebesgue integral

What are the advantages of the Daniel Integral over the Lebesgue integral and visa-versa? Heuristically speaking, I was wondering why this axiomatic operator is less popular besides the fact that it ...
1
vote
0answers
17 views

How does a Dirac delta function operate on a Fourier-Stieltjes integral?

Consider a stationary complex random function $\zeta(t)$ represented as a Fourier-Stieltjes integral $$\zeta(t) = \int_{-\infty}^{+\infty} e^{i\omega t}dA(\omega)$$ where $dA(\omega)$ is the random ...
0
votes
0answers
20 views

determining the fermi velocity via density of states

The problem is to determine the Fermi velocity for a fermion gas at absolute zero. the problem using integrating a function that looks like $$ v = \frac{4\pi V}{h^{3}} m^{3} \int_{0}^{\infty}{ ...
0
votes
0answers
21 views

Triple integral boundaries

If $W\subset \mathbb{R}^3$ is bounded by the planes $y+z=2, 2x=y, x=0, z=0$, what are the boundaries of $\int\int\int_W x dV$? How can I find the boundaries if I take $dV$ as $dydxdz$, $dxdzdy$ and ...
0
votes
1answer
26 views

Inverse Fourier transform of Gaussian

I need to calculate the Inverse Fourier Transform of this Gaussian function: $\frac{1}{\sqrt{2\pi}} exp(\frac{-k^2 \sigma^2}{2})$ where $\sigma > 0$, namely I have to calculate the following ...
2
votes
1answer
32 views

Evaluating an integral by dominated convergence theorem

I would like to know how to solve this two problems: a) $$ \lim_{n\to \infty}\int_0^n \left( 1-\frac{x}{n} \right)^{-n}\log{(2+\cos(x/n))} \, dx $$ b) $$ \lim_{n\to \infty}\int_0^{\infty} n e^{-nx} ...
2
votes
4answers
288 views

There is some strategy to solve an integral of this kind?

How to solve the integral $$\int\frac{\ln x}{\sqrt{1-x}}dx$$ and $$\int\sqrt{\frac{x}{x-1}}dx$$ I have no idea of how to deal with these integrals. It's the first integral I attempted. ...
1
vote
1answer
42 views

Integral involving CDF of a normal distribution

Can we evaluate the following integral ? $$\int_0^\infty x e^{-x^2} \Phi(ax+b)\,\mathrm dx$$ Here $\Phi(\cdot)$ is the cumulative probability distribution function of a standard normal random ...
2
votes
2answers
23 views

Finding the bounds of a solid for triple integrals

Ok, so I have an answer, most likely the wrong one. The question being asked is: Using polar coordinates find the volume of the solid bounded below by the $xy–plane$ and above by the surface $x^2 ...
1
vote
1answer
27 views

Integral of a funtion avoiding hypergeometric functions

I'm solving the following differential equation: $$uy''(u)+\gamma y'(u)+\frac{1}{u(1-u)}=0,~~\gamma=constant$$ For that, I transform this equation into a first orer one: $$uf'(u)+\gamma ...
-1
votes
1answer
39 views

What do these questions mean?

Can someone please explain what these questions mean in simpler terms? 1)If g is continuous on $[a,b]$ and for all $ x ∈ [a,b] $ we have $ g(x) ≥ 0 $ and also $ g(x_o) > 0$ for some $x_o ∈ ...