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 ...

learn more… | top users | synonyms (3)

0
votes
1answer
178 views
+100

Double obstructing wall problem, what is the optimal walk path and length?

Every day, you walk from point A to point B which are exactly $2$ miles apart straight line distance, however, each day, there is a $50$% chance of there being an obstructing wall perpendicular to the ...
4
votes
3answers
60 views

Exponential integration

I am working for an investment institution and we need to use upper partial moments. To evaluate them, I need to integrate $$ \int_a^\infty x^2 e^{-ax^2} dx $$ with $a>0$. I found this formula ...
0
votes
1answer
35 views

How to calculate $ \int_0^{\frac{ \pi }{2}} \frac{\sin^{1395}x}{\sin^{1395}x + \cos^{1395}x}\ dx $?

Solve the following integrals: $$ \int_0^{\frac{ \pi }{2}} \frac{\sin^{1395}x}{\sin^{1395}x + \cos^{1395}x}\ dx $$ And it does not follow a specific pattern.
4
votes
1answer
21 views

Mixed partials of the Beta function B$(a,b)$ at $(1,0^+)$

In this post M.N.C.E gave the equality below $$\frac{\partial ^{5}}{\partial a^{3}\partial b^{2}}\mathrm{B}\left ( 1,0^{+} \right )=\left [ \frac{1}{b}+O\left ( 1 \right ) \right ]\left [ \left ( ...
0
votes
1answer
23 views

Evaluate the double integral $\iint_D\sqrt{4-x^2-y^2}$ bounded by semi-circle

I would appreciate it if someone can help me solve this question, as I'm struggling to get its answer. Q: Evaluate the double integral $$\iint_D\sqrt{4-x^2-y^2}dxdy$$ bounded by semi-circle ...
1
vote
1answer
28 views

Line integral integrand

I have a doubt regarding integration of line integrals , in the books that I refer the integrand is usually parameterised to bring it in terms of a single variable. But I don't know why ...
1
vote
1answer
55 views

Seeking help with an error function Integral

I am trying to compute the following Integral $$ I = \int_{0}^\infty x \exp \left(-2 x \right) \operatorname{erf}\left(\frac{x}{t^{H}\sqrt[4]{2}}-\frac{t^H}{2^{3/4}}\right) \, dx $$ where ...
0
votes
0answers
17 views

Finding integral of product of Gaussian and error function

What is the answer for $$ \int_{y-b}^{y+b}exp(-\frac{(x-\mu)^2}{\sigma_1})erf(\frac{(\theta-x)}{\sigma_2})dx $$
0
votes
0answers
49 views

How this integral can be evaluated [on hold]

$$\ F(x;\alpha,\beta)=\alpha \beta \lambda \int_0^ \infty \frac {e^{tx} (1+ \lambda x)^{\alpha -1}}{[1- \beta + \beta (1+ \lambda x)^\alpha)]^{2}} \mathrm{d}x$$
1
vote
0answers
13 views

rearranging integral of cross product

I am given the integral \begin{gather} \int_V \hat{e}_z \times \vec{u} dV \end{gather} where $\hat{e}_z$ is the unit vector in the $z$ direction and $\vec{u}$ is a vector field. Can I pull ...
2
votes
1answer
22 views

Solving a separable integral equation: $y(x) = 1+\int_{1}^{x} \frac{y(t)dt}{t(t+1)}dt$

Solving integral equation. My answer is wrong. Where do I make a mistake? $$y(x) = 1+\int_{1}^{x} \frac{y(t)dt}{t(t+1)}dt $$ $$ y'(x) = \frac{d}{dx} \int_{1}^{x} \frac{y(t)dt}{t(t+1)}dt$$ $$ ...
7
votes
0answers
48 views

A version of Ampère's law

The most common proof that I have found of the fact that Ampère's law is entailed by the Biot-Savart law uses the fact that, if $\boldsymbol{J}:\mathbb{R}^3\to\mathbb{R}^3$, $\boldsymbol{J}\in ...
0
votes
0answers
25 views

Multiple Integral using Integration by Parts?

So basically I need to prove the following integral over the sphere: $x_1^2+...+x_n^2\leq R$ $$\int...\int 4\pi x_1^2...4\pi x_n^2dx_1...dx_n=\frac{(8\pi R^3)^N}{(3N)!}$$ Using the result of the ...
-1
votes
1answer
57 views

Riemann Sum proof: x^2 [closed]

How can I prove that $$\int _{a}^{b} x^2dx = \frac{b^3-a^3}{3}$$ using the definition of Riemann sum?
0
votes
0answers
26 views

Prove the property div(fG)= f*div(G)+G*grad(f)

Let f(x,y,z) be a function of three variables and G(x,y,z) be a vector field defined in 3D space. Prove the identity: div(fG)= fdiv(G)+Ggrad(f) I seem to keep getting it to equal 2div(fG) which is ...
4
votes
1answer
75 views

How to evaluate $\int_{0}^{\infty }\frac{e^{-x^{2}}}{\sqrt{t^{2}+x}}\mathrm{d}x$

How to evaluate the integral below $$\int_{0}^{\infty }\frac{e^{-x^{2}}}{\sqrt{t^{2}+x}}\mathrm{d}x~~~~~~(t>0)$$ The WolframAlpha gave me a horrible answer $$\frac{t}{2}e^{-\frac{t^{4}}{2}}\left \{ ...
1
vote
4answers
102 views

Integral: Trig Substitution?

An AP question asks: Compute $$\int_0^{1/3}\frac{9}{1+9t^2}\,dt$$ The only way to solve this is through trig substitution, correct? Since this is supposed to be an AB Calculus question, I ...
0
votes
0answers
19 views

On an estimation of a integral

I have the following function \begin{equation} S(x)=\int_{x_0}^x exp \left(-2 \int_{x_0}^y \frac{\beta(n-z)-a}{\beta(n-z)+a}dz \right)dy \end{equation} defined for $x \in [0, n+\frac{a}{b}]$ where ...
0
votes
0answers
10 views

Lagrangian method with objective function and constraints in expected value form.

Im reading a paper and over last two weeks I have been involved with a mathematical calculation. It is about maximizing the principal utility under uncertainty; max $\int G(x-s(x))f(x,a)dx $ , where ...
0
votes
1answer
43 views

how to solve this differential equation $(3(x^5)+3(x^2)(y^2))dx + (2(y^3)-2(x^3)y)dy = 0$

This is my first question here. I tried to solve this ODE. This is the Wolfram's answer but there's a step-by-step solution. :( Thanks
2
votes
1answer
59 views

Hellinger integral properties - proof of equivalence for infinite product measures

I'm trying to prove that: Let $(\mu_k)_{k=1}^{\infty}$ and $(\nu_k)_{k=1}^{\infty}$ be sequences of probability measures on $(\Omega_k, \mathcal{F}_k)$. Consider the product measures on ...
0
votes
1answer
27 views

Find $\int_{|z|=r}\frac{|dz|}{|z-a|^2}$ where $|a| \neq r$

Trying to find $$\int_{|z|=r}\frac{|dz|}{|z-a|^2}$$ where $|a| \neq r$. Was trying to use the maximum length estimate. For both case $|a|>r$ and $|a|<r$, I got the same answer zero.
0
votes
1answer
13 views

Bounds of a solid region triple integral

I am trying to find the integral of (e^y)dV, where D is the solid region bounded by planes y=1, z=0, y=x, y=-x, and z=y. I set the integral up as (e^y)dzdydz but I am unsure how to determine the ...
0
votes
0answers
31 views

Volume of an elipsoid using Gauss' Divergence Theorem

Question: Let $F= (0,0,z)$ be a vector field. Use Gauss' Divergence Theorem to calculate the volume of the ellipsoid $x^2+y^2+2z^2=1$ My attempt: $$r(a,b) = ...
0
votes
1answer
25 views

What is a puiseux series and what is wolfram-alpha doing with this antiderivative?

I asked wolfram alpha to compute the antiderivative of the function $x^x$. It gave me some really large confusing polynomial-esque thing called a puiseux series. However, from what I can gather on the ...
3
votes
3answers
61 views

Why are function spaces generally infinite dimensional

The other day, I was trying to explain some concepts in Fourier analysis and wavelets to my girlfriend (an electrical engineering student) and obviously, the concept of Lebesgue integration came up in ...
0
votes
1answer
26 views

Proof by induction, system of equations

We conjecture that there is a formula of the form $\sum_{j=1}^{n}{j^2} = an^3 + bn^2 + cn + d$ for all integers n ≥ 1 (3) (a) Assuming that such a formula is true, find the value of a, b, c, d. ...
1
vote
1answer
74 views

Why can a $u$ substitution not be used in this integral?

For the integral: $$\int \frac{v}{2 + v} dv.$$ I tried to use a $u$-substitution where $u = 2 + v$, rearranging to get $v = u - 2$ and $du = dv$. When I subbed this back in and integrated I got ...
1
vote
1answer
21 views

Generalization of the Beltrami identity to functionals with higher derivatives

Suppose that I have a functional $S[q]=\int_a^b L(t,q(t),q'(t))\,dt$. Such a functional is well-known to extremized by a choice of $q(t)$ satisfying the Euler-Lagrange equation ...
0
votes
1answer
76 views

Problem on divergence, rotation, flux

In $(x, y, z)$ space is considered the vector field $V(x,y,z)=(y^2 z, yx^2, ye^y)$ solid spatial region $Ω$ is given by the parameterization: $\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] ...
0
votes
1answer
11 views

Integration - $p\int_\frac{S}{a}^T(0-at)dt$

I have a formula: $$p\int_\frac{S}{a}^T(0-at)dt$$ I did the following: $$p(\int_\frac{S}{a}^T0dt-\int_\frac{S}{a}^Tatdt)$$ $$=p(0-\frac{aT^2}{2}-\frac{\frac{S^2}{a}}{2})$$ ...
1
vote
2answers
70 views

Can the limit of averages of $f(1),f(2),\dots, f(n)$ be expressed as an integral?

If $\int_0^1 f(x) dx$ exists then, of course, $$ \lim_{n\to\infty} \frac{f(\frac{1}{n})+f(\frac{2}{n})+\ldots+f(\frac{n}{n})}{n} = \int_0^1 f(x) dx. $$ I would like to know is there a similar formula ...
2
votes
2answers
747 views

Find the second derivative of a double integral

Problem: Find $F''(\pi)$ if $$ F(x) = \int_x^{\sin x} \left(\int_0^{\sin t} (1+u^4)^{0.5} \,du\right)\,dt $$ Context I don't know how to integrate the inner part. What to do to the integral inside ...
5
votes
0answers
547 views
+50

Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
2
votes
2answers
65 views

Find the area of this quadrilateral using double integrals

Evaluate this double integral: $\int \int _{R} x dx dy$ where $R$ is the area of a quadrilateral of vertex $(0,-1),(5,-1),(3,1),(2,1).$ When i do the double integral i get the answer $15$. In the ...
2
votes
0answers
30 views

Existence of a measurable function

The question is Let $\theta>0$ fixed. Is there a measurable function $h:[0,1]\to\mathbb{R}$ such that $\int_0^1h(x)x^{\theta-1}dx=1$ and $h$ is independent of $\theta$? Could you check my ...
1
vote
0answers
32 views

Remaining volume percentage of a sphere

Circular tunnel of radius $r$ is punctured through the center of a homogeneous sphere of radius $R, (r<R)$. What percentage of a sphere is lost? What should be the value of $r$ such that the sphere ...
0
votes
1answer
24 views

Compute the line integral where $C$ is an arbitrary smooth path from $(0,0,1)$ to $(\pi,\pi,0)$

Compute $∫_C −e^y\sin(x) dx + e^y\cos(x) dy + dz$, where $C$ is an arbitrary smooth path from $(0, 0, 1)$ to $(π, π, 0)$. Make sure to check you satisfy the hypotheses of any theorems you use. I went ...
0
votes
1answer
28 views

Simple Integration Identity

Prove, by sketching the region of integration and interchanging the order of integration, that $$ \int_a^x \int_a^{\xi} f(s) ds d\xi = \int_a^x (x-s)f(s)ds $$ This is probably quite simple ...
1
vote
3answers
61 views

Combining error terms in Simpson's rule

My numerical analysis textbook (Burden and Faires) derives Simpson's rule as $$\begin{align} \int_{x_0}^{x_2}f(x)\,dx&=2hf(x_1)+\frac{h^3}{3}f''(x_1)+\frac{h^5}{60}f^{(4)}(\xi_1) ...
89
votes
24answers
8k views

Is there any integral for the Golden Ratio?

This is a curiosity. I was wondering about math important/famous constants, like $e$, $\pi$, $\gamma$ and obviously $\phi$. The first three ones are really well known, and there are lots of integrals ...
0
votes
0answers
11 views

Help understanding the proof of Trace Theorem given in Evans

I need help to understand the proof of the Trace Theorem given in Evans L.C. Partial differential equations (AMS, 1997): Asume $U$ is a bounded open set and that $\partial U$ is $C^1$. Then there ...
0
votes
0answers
24 views

De Moivre's population recursive formula

For a de Moivre's (i mean, that follows de Moivre's law) population with unknown parameter $\omega$, knowing $\dot{e}_{x+1}$ is given. From the recursive formula find $\dot{e}_x$ I am not certain ...
0
votes
1answer
24 views

Hessian under integral sign

If I have a function $f(a,b)$ s.t. $f(a,b)=\int_{I}{f(x,a,b)dx}$, how would I be able to characterize the point $(a^{*},b^{*})$ where $f(a^{*},b^{*})$ is an extremum? Would the Hessian simply be ...
0
votes
1answer
16 views

How to find parametric equation between two points in line integral?

[In this example how can we find parametric equations of x and y.] [1] [question]: http://i.stack.imgur.com/lTOnW.png [1] [Solution]: http://i.stack.imgur.com/l8ao7.jpg
1
vote
1answer
76 views

Friend claims $\int_0^\infty\sum_{n=0}^{\infty}\frac{x^n}{2^{(n+1)^sx^{n+1}}+1}dx=\zeta(s+1)$?

My friend is making another claim on another integral! Can anybody verify it? Or his is mocking on me? Valid for all $s\ge1$ ...
4
votes
2answers
129 views
+50

Let $f:[-1,1]\to \mathbb{R}$ diferentiable, with $f'$ integrable, such that $\frac{\int_{-1}^{1}e^xf(x)dx}{f(1)-f(-1)}=2(e+e^{-1})$.

Let $f:[-1,1]\to \mathbb{R}$ diferentiable, with $f'$ integrable, such that $$\frac{\int_{-1}^{1}e^xf(x)dx}{f(1)-f(-1)}=2(e+e^{-1})$$ Prove that there exists $c\in (-1,1)$ such that ...
0
votes
2answers
62 views

Find $\int (\sin^2 x - 2\cos^2 x)\,dx$

Find $$\int (\sin^2 x - 2\cos^2 x)\,dx$$ $$=\frac{1}3 -\cos^3x - \frac{1}4 \sin^3 x$$ This is of course not the right answer which is $$-\frac{3}4 \sin 2x - \frac{1}2 x + C$$
0
votes
2answers
32 views

Integral path between 2 points

so I need your help calculating the next inegral: Calculate the integral $$\int(10x^4-2xy^3)dx -3x^2y^2dy$$ at the path $$x^4-6xy^3=4y^2$$ between the points $O(0,0)$ to $A(2,1)$ please explain me ...
3
votes
1answer
48 views

A certain multidimensional integral.

Consider a following multidimensional integral: \begin{equation} \bar{I}^{(t_0,t)}_p := \int\limits_{t_0 \le \xi_0 \le \cdots \le \xi_{p-1} \le t} \prod\limits_{j=0}^p (\xi_{j-1}-\xi_j) \cdot ...