All aspects of integration, including the definition of the integral and computing indefinite integrals (antiderivatives).
12
votes
0answers
649 views
Computing ${\int\limits_{0}^{\chi}\int\limits_{0}^{\chi}\int\limits_{0}^{2\pi}\sqrt{1- \cdots} \, d\phi \, d\theta_1 \, d\theta_2}$?
This question led me to this integral I can not solve:
$$
...
10
votes
0answers
148 views
definite and indefinite sums and integrals
It just occurred to me that I tend to think of integrals primarily as indefinite integrals and sums primarily as definite sums. That is, when I see a definite integral, my first approach at solving it ...
8
votes
0answers
291 views
a way of evaluating integrals without doing anything?
The user known as sos440 posted this:
$$\begin{align*} \sum_{n=0}^\infty \frac{r^n}{n!} \int_0^\infty x^n e^{-x} \; dx & = \int_{0}^\infty \sum_{n=0}^\infty \frac{(rx)^n}{n!} e^{-x} \; dx = ...
7
votes
0answers
172 views
Evaluating $\int{ \frac{\arctan\sqrt{n^{2}-1}}{\sqrt{n^{2}+n}}} dn$
How to integrate?
$$\int{ \frac{\arctan\sqrt{n^{2}-1}}{\sqrt{n^{2}+n}}} dn$$
I have no idea how to do it.
Tried to get some information from wiki, but its too hard :|
7
votes
0answers
313 views
Cauchy-Formula for Repeated Lebesgue-Integration
Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given.
Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
6
votes
0answers
59 views
Asymptotic Expansion of an Oscillating Integral
Let $g(x):\mathbb{R}_{\geq0}\rightarrow\mathbb{R}$ be real analytic s.t. $g(0)\neq 0$ and $g(x)=O(x^{-2})$ as $x\rightarrow\infty$.
What is the leading order in $\lambda$ as $\lambda\rightarrow 0$ of ...
6
votes
0answers
153 views
Computing the volume of a region on the unit $n$-sphere
I would like to compute the surface volume of a region on the unit $n-1$-sphere:
$$x_1^2 + \dots + x_i^2 + \dots + x_n^2 = 1,$$
bounded by an ellipsoid
$$a_1x_1^2 + \dots + a_ix_i^2 + \dots + ...
5
votes
0answers
64 views
What is the relation of $\int f dx^1\wedge dx^2\wedge …\wedge dx^n=\int f dx^1…dx^n$
In a book "calculus on manifolds" it is defined that $\int f dx^1\wedge dx^2\wedge ...\wedge dx^n=\int f dx^1...dx^n$ but how it is possible the relate the integrand of a multilinear function ...
5
votes
0answers
222 views
Is this question solvable? $2$ non-linear equations and the proof that the solution is unique (with asymmetric bounty option)
As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
5
votes
0answers
98 views
Nontrivial trivial integrals
Consider two propositions in geometry:
Circumscribe a right circular cylinder about a sphere. The surface area of the cylinder between any two planes orthogonal to the cylinder's axis equals the ...
5
votes
0answers
125 views
An integration to first order
I am having some trouble evaluating an integral -- involving taking an approximation. It would be great if someone could help me.
I wish to evaluate
$$\int_0^\pi {\cos\theta\cos \left[\omega ...
5
votes
0answers
51 views
Various integration theories
Could anyone briefly explain, or point me towards a resource explaining, the main differences between the main integration theories, namely:
Riemann Integration
Riemann-Stieltjes Integration
...
5
votes
0answers
189 views
To determine if$f^{-1}(x)$ is periodic function or not? $f(x)=\int_1^{x} \frac{1}{\sqrt[m]{P(t)}}\;dt$
$$f(x)=\int_1^{x} \frac{1}{\sqrt[m]{P(t)}}\;dt$$
$P(x)$ is polynomial with degree $n$.
$m$ is an positive integer and $m>1$
What is the algoritm to determine $f^{-1}(x)$ is periodic function ...
5
votes
0answers
115 views
Evaluting $ \int_0^{\infty}\frac{v}{\sqrt{v + c}}e^{-\frac{y^2}{2(v + c)} - \frac{(u-v)^2}{u^2v}}dv$
While working on mixture (variance) of normal distribution and keep running into these two integrals
$$ \int_0^{\infty}\dfrac{v}{\sqrt{v + c}}e^{-\dfrac{y^2}{2(v + c)} - \dfrac{(u-v)^2}{u^2v}}dv,$$
...
4
votes
0answers
51 views
Integration by parts of a normalized function - [copied from Physics.SE]
By using integration by parts, I need to show for $$A = \frac{\mathrm d}{\mathrm dx} + \tanh x, \qquad A^{\dagger} = - \frac{\mathrm d}{\mathrm dx} + \tanh x,$$ that
...
4
votes
0answers
59 views
How can I Create an integral that can only be evaluated via complex contour integration?
In Richard Feynman's book, Surely You're Joking Mr. Feynman!, he says:
One time I boasted, "I can do by other methods any integral anybody else needs contour integration to do."
So Paul ...
4
votes
0answers
73 views
Contour integration of $\int_{-\infty}^{\infty}e^{iax^2}dx$
Consider the following integral:
$$\int_{-\infty}^{\infty}e^{iax^2}dx$$
Here I believe we have to consider the two cases when $a<0$ and $a>0$, as they need different contours. For $a>0$ ...
4
votes
0answers
63 views
Stokes' Theorem Example
I am reading Wade's Introduction to Analysis. One of the exercises is to show that
$$
\int_{\partial M}\sum_{k=1}^n \, dx_1dx_2\cdots \hat{dx_i}\cdots dx_n
$$
is equal to the volume of $M$ if $n$ is ...
4
votes
0answers
83 views
Gaussian Integral with non-polynomial exponent
I am currently trying to evaluate this Integral:
$$\int\limits_{u_0}^{u_1} \exp\left[-\angle(H(u),N)^2\right]du$$
Where ...
4
votes
0answers
62 views
Dominated convergence on $e^{-n^2 t} t^{s/2-1}$
I am trying to apply the Dominated Convergence Theorem to show that
$$\sum_{n\ge 1} \int_0^1 e^{-n^2 t} t^{s/2-1}dt= \int_0^1 \sum_{n\ge 1}e^{-n^2 t} t^{s/2-1}dt$$ as soon as $s>1$.
I've ...
4
votes
0answers
143 views
Evaluate the integral by converting to polar coordinate
$$ \int^{\pi/2}_{\pi/4} \int^{\sqrt{2-y^2}}_y 3(x-y) dx dy$$
I attempted the following:
$$ \int_{\pi/4}^{\pi/2} \int_{0}^{1} 3r^2 (\cos\theta - \sin\theta) dr d\theta $$
which is wrong apparently. ...
4
votes
0answers
138 views
Recurrence relation for the integral, $ I_n=\int\frac{dx}{(1+x^2)^n} $
Express recurrence relation of the integral
$$
I_n=\int\frac{dx}{(1+x^2)^n}
$$
[My Answer]
$$
I_n = \int\frac{1+x^2}{(1+x^2)^n}dx-\int\frac{x^2}{(1+x^2)^n}dx
$$
$$
I_n=I_{n-1}-\int ...
3
votes
0answers
30 views
What is $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$?
I want to compute the following integral:
$\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$
with $\alpha, \beta, c$ real constants, and $\alpha>0,\beta=0$.
...
3
votes
0answers
152 views
+100
About Henstock integrable vector-valued function
In what follows, $X$ is a Hausdorff locally convex topological vector space over the reals whose topology is generated by a family $P$ of all continuous seminorms on $X$. We consider the following ...
3
votes
0answers
56 views
Simplify the integral with error function
$\newcommand{\erf}{\operatorname{erf}}$
I have the following integral and I need to simplify the solution. I have written first two steps. I don't know what is the value of
$$
\erf(\infty)
$$
I ...
3
votes
0answers
101 views
An absolutely continuous cumulative distribution function that fails to have a Riemann-integrable pdf.
We know that if a real-valued random variable $ X $ on a probability space has an absolutely continuous cumulative distribution function (cdf) $ F $, then $ X $ possesses a probability density ...
3
votes
0answers
208 views
Riemann integral vs Lebesgue integral
Let $f$ be analytic on a domain $\Omega$ of the complex plane, such that the closed disc $\overline{D(0,R)}$ is contained in $\Omega$. What is the difference between
$$ \int_{D(0,R)}|f(w)|dm(w)$$
and
...
3
votes
0answers
89 views
Hazard Rate Probability HW Question
I am working on a homework problem below from Pitman's Probability book:
Suppose the failure rate is $\lambda$($t$) = $at$ $+$ $b$ for $t$ $\geq 0$.
The problem asks to find the formula for the ...
3
votes
0answers
56 views
why we cannot integrate on a nonorientable manifold?
I feel it rather weird that there is a notion of integration when you glue a patch of paper to get a surface of cylinder while there is not a suitable notion when you glue it differently to get a ...
3
votes
0answers
51 views
Can I solve for a unique integral kernel?
Consider, for $\mathbf{v},\mathbf{w} \in \mathbb{R^3}$,
$$ f(\mathbf{w}) := \int K(\mathbf{w,\mathbf{v}}) g(\mathbf{v}) \, d\mathbf{v} \, .$$
Is it possible to solve for the integral kernel, ...
3
votes
0answers
258 views
Integration of nontrivial trigonometric functions
First an example which I know how to solve. If we have the following integral
$$\int_{-\pi}^{\pi}\frac{1}{1+3~\cos^2(t)}dt$$
there is a very practical way to evaluate it by interpreting it as some ...
3
votes
0answers
120 views
Lebesgue Integration fundamental questions
My question involves the definition of the Lebesgue integral. Most colloquial definitions I've read follow (2), in that f*(t) is the "length" of one of the horizontal rectangles and dt is the ...
3
votes
0answers
150 views
Generalized Change of Variables Theorem?
Is there a generalized form of the differentiable change of variables theorem for Lebesgue integrals? That is, if we consider the well known change of variables theorem: If $\phi : X \rightarrow X$ is ...
3
votes
0answers
121 views
Equivalence of integrals
Let $x_1, \ldots, x_n$ be vectors in the normed space $(X, \|\cdot\|)$.
Let $\mu$ be the Lebesgue measure on the cube $[-1,1]^n$. Denote vectors in $[-1,1]^n$ by $y=(y_1, \ldots, y_n)$.
Are the ...
3
votes
0answers
116 views
Integral with exp, ln and arccosh
I would be interested in any clue on how to carry out the following integral
$$\int_1^\infty \cosh^{-1}(x) \ln(x^2-1) \exp \left(- \frac{x}{T} \right) dx $$
I have tried integration by part but it ...
3
votes
0answers
55 views
How to approach an integral over $g(\cos(t))$ from $0$ to $2\pi$, where $g(x)$ is nasty?
For notational convenience, let $f(t) = a^2 + 2 a b \cdot \cos(t) + b^2$, where $a,b$ are both positive real constants and $t$ will be the integrand of the integral, which is supposed to be carried ...
3
votes
0answers
152 views
Integrating a product of exponential and complementary error function with square-root of variable in the denominator
I need to evaluate
\begin{equation}
\int_a^\infty \mathrm{erfc}\left( \frac{b}{\sqrt{c\cdot h}} \right) e^{-d\cdot h} dh
\end{equation}
where $\mathrm{erfc}(s) = \frac{2}{\sqrt{\pi}} ...
3
votes
0answers
186 views
Taking derivative below an integral
I am trying to solve the following question:
If $t>0$, then
\begin{align*}
\int_{0}^{+\infty} e^{-tx} \; dx = \frac{1}{t}
\end{align*}
Moreover, if $t \geq a > 0$, then $e^{-tx} \leq ...
3
votes
0answers
93 views
What is the Dunford Integral and why is it useful?
Wikipedia defines the Pettis Integral for Banach space valued functions on a measure space by duality. Apparently there is a Dunford integral which specializes to the Pettis integral. What is its ...
3
votes
0answers
75 views
Asymptotics of Riemann-Lebesgue type integral
How to show that for $u \in L_{\mathbb{C}}^2$ and $a>0$,
$$\int_0^a u(t) \sin{\sqrt{\lambda}t} \,dt = o(e^{|Im\sqrt{\lambda}|a}),\text{ as } |\lambda| \rightarrow \infty$$
Note that $\lambda$ ...
3
votes
0answers
137 views
Evaluating $\int\limits_{-\infty}^\infty {\exp(iax)\over1+ix}dx$
How does one evaluate the integral $\int\limits_{-\infty}^\infty {\exp(iax)\over1+ix}dx$? I tried Wolfram Alpha, but it just says "computation timed out"... I tried the indefinite integral and got an ...
3
votes
0answers
54 views
Integrability of $\frac{k-1}{2(1+|x|^k)}$
Is the following function known to be integrable? It is supposed to be a probability density function, i.e., integrates to one. However, it leaves the online Mathematica integrator stumped:
...
3
votes
0answers
129 views
Change of variables in line integral with abs. value
Let $\gamma : I \rightarrow \mathbb C$ be a path. Let $g: \mathbb C \rightarrow \mathbb C$ be a biholomorphic map. Let $f$ be a holomorphic function. Consider the integral
$$ \int_{g\circ \gamma} ...
2
votes
0answers
32 views
Closed curves question
Can you give me some help on the following problem?
Given two closed curves $\alpha, \beta : \mapsto \mathbb{R}^3$ we define $\phi_{\alpha \beta}: I^2 \mapsto \mathbb{R}^3$ as $\phi_{\alpha \beta} ...
2
votes
0answers
38 views
Interchanging the limiting operations
How to remember the conditions for interchanging the limiting operations , for example between limits and integrals or integrals and sums or derivation of any order and integrals, i mean every one of ...
2
votes
0answers
60 views
A photon in expanding Universe (a snail on a tree)
I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of ...
2
votes
0answers
88 views
Injectivity in the Change of Variables Formula
I badly need to apply the change of variables
\begin{array}{ll}
u= t+s \\
v= t^2+s^2;
\end{array}
i.e. $(u,v)= T(t,s)= (t+s,t^2+s^2)$,
to an integral;
however, ...
2
votes
0answers
36 views
computing a difficult integral using software
I'd like to compute the following integral. I've tried SAGE but it just runs for 15 minutes then stops.. not sure what that means. If anyone wants to take a crack with mathematica or anything, please ...
2
votes
0answers
37 views
Intuitive understanding of integral of vector valued functions
Today in class we were introducing complex line integrals. And that got me thinking, I don't know of a good interpretation for integrals of functions from $\mathbb{R}$ to $\mathbb{R}^2$ or ...
2
votes
0answers
60 views
Prove Heisenberg uncertainty principle (measure and integration theory)
Here is a question in measure and integration theory,
Let $f$ be a continuously differentiable complex function on $\mathbf{R}$ s.t. the functions $x \mapsto xf(x)$ and $f'$ are in ...
