1
vote
2answers
26 views

Double Integral Over a General Region

Evaluate the $\iint_R \sin\left(\frac{(x+y)}{2}\right)\cos\left(\frac{(x-y)}{2}\right) dA$ where R is a triangle with vertices $(0,0), (2,0), (1,1)$ using $u = \frac{(x+y)}{2}$ and $v = ...
2
votes
1answer
26 views

How do I set up a triple integral with vague parameters in the z coordinate direction?

The problem asks to find the volume of the solid bounded by $z = 16xy,~ z \geq 0,~ 0 \leq x \leq 5,~ 0 \leq y \leq 4$. But I am having trouble setting up the integral for the $z$ parameters. Any help ...
2
votes
2answers
28 views

convergence of a sequence

I'm reading a paper, for proving a claim it defines $$ f_n(x) = \dfrac{(rx-x^2)^n}{n!} $$ when $ r = \frac{a}{b} $ is a rational, and $ I_n = \int^r_0 f_n(x) \cdot \sin x \cdot dx $ , and then it says ...
3
votes
2answers
55 views

Proving $\int_0^1 \frac{\mathrm{d}x}{1-\lfloor \log_2(1-x)\rfloor} = 2 \log 2 - 1$.

By testing in maple I found that $$ \int_0^1 \frac{\mathrm{d}x}{1-\lfloor \log_2(1-x)\rfloor} = 2 \log 2 - 1 $$ Does there exists a proof for this? I tried rewriting it as an series but no luck ...
2
votes
0answers
23 views

Prove that $I = \int_0^{m(m+1} y_n(x)\,\mathrm{d}x$ converges and $I \in \mathbb{Q}$.

My problem is stated as follows Let $y_0(x) = x, \ \: y_1(x) = \sqrt{x}, \ \: y_{n+1}(x) = \sqrt{y_n(x) +x\,} \ $. Now define $ \displaystyle \hspace{3cm} I_n = \int_0^k ...
1
vote
0answers
28 views

How to establish the equivalence of these two statements about integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a, b]$. Then we have $$ \int_{ka}^{kb} s\left(\frac{x}{k}\right) \ dx = k \int_a^b s(x) \ dx $$ for every $k ...
0
votes
0answers
23 views

How to establish this equivalence for integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a,b]$. Then we have $$\int_{a}^{b} s(x) \ dx = \int_{a+c}^{b+c} s(x-c) \ dx.$$ Second Statement: Let $s$ be ...
1
vote
1answer
33 views

Integrating the delta function

So i was trying to figure out how the delta function is integrated and i got lost trying to figure out how they reached the last step, i tried integration by parts but got stuck ...
1
vote
2answers
48 views

Integration and differentiation complicated equation

What does this mean? $$ \frac{d}{dx}\left.\right|_{x=\pi} \int_{t=0}^x \frac{\cos (3t)}{\sqrt{1+t}} dt $$ Do I have to differentiate after I find the integral? Can anybody help me solve it? It's ...
0
votes
1answer
36 views

Example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable?

Can anyone come up with an example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable? Thanks.
1
vote
3answers
26 views

Growth restriction for nonnegative, continuous functions whose integrals on $\mathbb{R}$ are bounded

When we have a nonnegative, continuous function $f(x)$ whose integral over all real numbers $\mathbb{R}$ is bounded, like: $$\int_{-\infty}^{\infty}f(x)dx = A< \infty $$ with $A \in \mathbb{R}$ ...
1
vote
0answers
25 views

Contour integral: different answers with different contours

Good day to everyone. I have a following contour integral problem. I have to find a solution for the integral $$\underset{\gamma_r }{\oint }\frac{e^{\lambda s} }{(1-s) s^{a-b} \left(s-\theta ...
1
vote
0answers
28 views

Solving indefinite integrals gives multiple answers. Are all those answers correct?

While solving problems on indefinite integrals many a times I get answers which are different from those given in my text book's answer keys page. I then verify my solution steps to ensure that even ...
-1
votes
1answer
15 views

Finding the equation of displacement

How to find equation of displacement when you know the equation of velocity. And the finding the constant when you know that at t=0 displacement is 2?
1
vote
0answers
42 views

Question on integral

I need a confirmation and answer about the following problem: If we have $ g(x)=\ln x+{ x }^{ -1/2 }{ 1 }_{ x\le 1 } $ I'm trying to determine $ \int _{ 0 }^{ +\infty }{ \ln x+{ x }^{ -1/2 }{ 1 }_{ ...
0
votes
0answers
25 views

Demand integral not a function of variable

I'm pretty sure this is not possible but I have to ask: Given a definite integral $I=\int_{x_a}^{x_b} dxf(x,y)$, in general we have $I=I(y)$; that $I$ is a function of $y$. Is there any other way to ...
2
votes
0answers
18 views

Problem with volume integral of a scalar function

I have difficulties in integrating this scalar function over the assigned volume. Let $D=\{(x,y,z):(x-2z)^2+(y-x)^2+(x+z)^2\le4,\,0\le x+y+z\le1\}$ Calculate $\int_D z\,dxdydz$
-1
votes
0answers
30 views

Prove integration formula for $\int\frac{dx}{({a^2+x^2})^{-3/2}}$? [on hold]

Prove integration formula for $\int\frac{dx}{({a^2+x^2})^{-3/2}}$? $\int\frac{dx}{({a^2+x^2})^{-3/2}}= \frac{xdx}{a^2\sqrt {a^2+x^2}}$
-4
votes
2answers
52 views

Prove integration formula for integral $\int_{}^{} \! \frac{1}{\sqrt{(x^2+a^2)}} \, dx$? [on hold]

Prove integration formula for $\int_{}^{} \! \frac{1}{\sqrt{(x^2+a^2)}} \, dx$ ? $\int_{}^{} \! \frac{1}{\sqrt{(x^2+a^2)}} \, dx= ln(x+\sqrt{(x^2+a^2)})$
-6
votes
1answer
44 views

Prove integration formula for integral $\frac{1}{\sqrt {a^2-x^2}}$? [on hold]

Prove integration formula for integral $\frac{1}{\sqrt {a^2-x^2}}$? $\int\frac{1}{\sqrt {a^2-x^2}}= \arcsin( x/a) +c$
0
votes
1answer
21 views

Problem with surface integral of a scalar function

I have difficulties in integrating this scalar function over the assigned surface. I've tried with the standard method but I always obtain an integral which I don't think can be solved. Let ...
1
vote
0answers
25 views

numerical solution of integral equation

Consider the basic type of integral equation. In particular, a volterra integral equation of the first kind. That is, we have the following integral equation $$\int_a^xf(s)g(s,x)~ds=h(x)$$ where $h$ ...
0
votes
0answers
20 views

Solving an integral equation in general

I have an integral equation such that $$\int_t^Tf(s)g(s,t)~ds=h(t)$$ where $g$ and $h$ is given. we want to know function $f$ explicitly. As I know, this type of question is about the integral ...
1
vote
0answers
31 views

How to fill in these steps to evaluate this Gaussian integral?

As a part of a much bigger problem, I came across this integral $$\int_{-\infty}^{\infty}\ln(|x|)\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}dx$$ which represents ...
6
votes
2answers
127 views

Integral: $\int_0^{\infty} \cos\left(\frac{a^2}{x^2}-b^2x^2\right)\,dx$for $a,b>0$

I tried this: $$\int_0^{\infty} \cos\left(\frac{a^2}{x^2}-b^2x^2\right)\,dx=\Re\left(\int_0^{\infty} e^{-ib^2x^2+ia^2/x^2}\,dx\right)=\Re\left(\int_0^{\infty} ...
1
vote
0answers
24 views

A calculus problem regarding mass

I am reviewing calculus and working on the following problem. In $\mathbb{R}^{3}$, the density function is given by $\mu(x, y, z) = |z|$ and if the region $R$ is given by $R : 2 \leq x^{2} + y^{2} + ...
0
votes
1answer
48 views

Which Riemann integrable functions have all lower sums equal?

From Spivak's Calculus, 4th edition, problem 13-11(d): Which (Riemann) integrable functions have the property that all lower sums are equal? (Bear in mind that one such function is $f(x)=0$ for ...
0
votes
1answer
52 views

Anti derivative notation [duplicate]

$F$ is an anti derivative of $f$. $$\int f(x) dx = F(x)+C$$ Can you tell me why there is '$dx$' in the LHS?
0
votes
1answer
19 views

How to identify continuity or discontinuity of an [Definite] integral?

How can I figure out whether an improper integral converges based on the discontinuities in the integrand? For instance, these two both have discontinuities within the intervals of integration, and ...
0
votes
1answer
19 views

How rigorous is multiplying both sides of an eqaution for the differential of a function?

I have to solve this equation: $$ -C_0 f + \frac{1}{2}f^2 +\frac{d^2 f}{d X^2}=A $$ where $C_0$ and $A$ are two real nonzero constant; $f:\mathcal{R}\to \mathcal{R}$ I have seen that the person who ...
0
votes
1answer
19 views

Expectation of exponential of Brownian motion

I want to compute the following expectation: $\mathbb{E}[\int_0^\infty-e^{-\mu t+\sigma W_t}dt]$ where $W_t$ is a brownian motion, $\mu$ and $\sigma$ constant. I am already stuck at computing the ...
3
votes
2answers
58 views

How to calculate $\int_{\partial B_2(0)}\frac{2z^2+7z+11}{z^3+4z^2-z-4}\;dz$?

I want to calculate $$\displaystyle\int_{\partial B_2(0)}\underbrace{\frac{2z^2+7z+11}{z^3+4z^2-z-4}}_{=:f(z)}\;dz\tag{0}$$ Partial fraction decomposition yields ...
0
votes
1answer
39 views

A question about properties of integrals

Suppose g is differentiable with $g'(x)<0$ for all $x<1$, and $g'(x)>0$ for all $x>1$, and suppose $g(1)=0$. Now let $G(x)=\int_0^x g(t)dt$. Prove that G(x) is an increasing function (this ...
0
votes
1answer
23 views

Gaussian integral with offset, and other cases

Consider the Gaussian Integral $$ \int_{-\infty}^{\infty} e^{-x^2} \ dx = \sqrt{\pi}$$ Numerically, it seems that for any arbitrary imaginary offset, ki, $$\int_{ki-\infty}^{ki+\infty} e^{-x^2} \ dx ...
1
vote
1answer
22 views

Choosing which technique to use to solve multiple integrals?

There are at least two ways to solve double integrals. One way is to use interated integrals, based on Fubini's, or similar theorems. The other way is to reduce the double integral to a line (curve) ...
0
votes
0answers
23 views

Requirements for integration by parts

In order to use the integration by parts formula for functions of several variables $$\int_{\Omega} \nabla u\cdot v d \Omega = \int_{\partial \Omega}(u(v \cdot \nu))d \Omega - \int_{\Omega}u\nabla ...
0
votes
0answers
21 views

Integration on Manifold

I am beginning my studies on integration on manifolds and i have some theorical questions. First, in all books that I saw they says that the singular p - simplex (or p - cube) are continuous mapping ...
1
vote
0answers
996 views

Is $ \lim_{x \to 0} G(e^x) x = \sum_{r=1}^\infty \mu(r) \int_{-\infty}^\infty \frac{G(z^r) dz}{z} $?

I recently derived a formula and was wondering if it was correct or already existing? We define: $$ \sum_{r=1}^\infty \mu(r) G(x^r) = g(x) $$ Where $ \mu(r) $ is the Mobius function. G(x) is an ...
0
votes
0answers
25 views

How to calculate $\int_{-\infty}^\infty e^{-t^2/2}\cos2t\ dt$ using Cauchy's integral theorem? [duplicate]

I need a hint. Where do I start if I want to calculate $$\int_{-\infty}^\infty e^{-t^2/2}\cos2t\ dt$$ using Cauchy's integral theorem?
0
votes
0answers
15 views

Quadrature methods: even order?

I noticed that all quadrature methods I know (Newton-Cotes and Gaussian quadrature) have always even order in the sense that a quadrature method is of order $n$, if all polynomials of degree $n-1$ are ...
1
vote
1answer
61 views

How to calculate an integral

I wonder how the integral $$\int_{-1}^1 \! \int_0^{\sqrt{1-x^2}} \! \int_0^{\sqrt{1-y^2-x^2}} \! 1 \, dz \, dy \, dx $$ Any ideas?
3
votes
0answers
40 views

On integration of a Gaussian-like function over a region $g(\mathbf{x})\leq 1$

Let $X$ be a random variable which follows an $n$-dimensional Gaussian distribution with mean vector $\mu\in\mathbb{R}^n$ and covariance matrix the symmetric positive definite $n\times n$ matrix ...
3
votes
1answer
49 views

Integration to Gamma Function?

I need to show that $$\int\limits_{0}^{\infty}\theta^{-\tau(\alpha_1 + \alpha_2) - 1}e^{-\theta^{-\tau}\left(y^{\tau} + \delta^{\tau}\right)}\text{ d}\theta = \dfrac{\Gamma\left(\alpha_1 + ...
0
votes
0answers
13 views

Probability density function of an element

How to find the probability density function of $x_m\left(1\le m\le n\right)$ from joint density function, $p_X\left(x_1,x_2,\cdots,x_n\right)$, of $n$ random variables which satisfy following ...
1
vote
0answers
26 views

Question about a particular estimate in Riemannian geometry.

I have been studying the book Some Nonlinear Problems In Riemannian Geometry - Thierry Aubin. On page $46$ he begins the proof of the Sobolev imbedding theorem to manifolds. The proof is divided in ...
0
votes
0answers
33 views

Integration: Countable Additive Measure?

When considering Bochner's theory of integration one notices that having a countable additive measure rather than merely additive measure is not important, or do I miss something?
1
vote
0answers
29 views
+50

Numerical integration of $\sin(p_{m})$ and $\cos(p_{m})$ for a polynomial $p_{m}$

I was wondering if anyone knew about any numerical methods specifically designed for integrating functions of the form $\sin(p_{m})$ and $\cos(p_{m})$ where $p_{m}$ is a polynomial of degree $m$. I ...
3
votes
0answers
38 views

2D Integral of Bessel Function and Gaussians

I've run into the following integral, and I'm not sure how to evaluate it. $$F(k)=\int ...
0
votes
1answer
46 views

Resources for learning integral calculations

I am willing to learn about integrals . So i wonder is there any systematic book about the topic that goes progressively in difficulty and complexity . My current level is about knowing the basic ...
1
vote
1answer
43 views

Haar measure on $G \times G$, where $G$ is compact

Let $G$ be a compact group. Let $\mu'$ and $\mu$ be the Haar measure on $G \times G$ and $G$, respectively, and further such that $\mu'(G \times G) = 1$ and $\mu(G)=1$. Does it follow that $\mu' = \mu ...