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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Integrating $\int \sin^n{x} \ dx$

I am working on trying to solve this problem: Prove: $\int \sin^n{x} \ dx = -\frac{1}{n} \cos{x} \cdot x \ \sin^{n - 1}{x} + \frac{n - 1}{n} \int \sin^{n - 2}{x} \ dx$ Here are the steps that I ...
4
votes
1answer
276 views

Is My Solution on Integration by Parts Correct?

For $x>0$ let $\ f(x) = \int_0^\infty e^{-t-x^2⁄t} t^{-1/2}dt $ the question wants us to show that $\ f(x) = x \int_0^\infty e^{-t-x^2⁄t} t^{-3/2}dt $ by using substitution. However I do not ...
4
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1answer
161 views

Is line integration a generalization of the definite integral in $\mathbb{R}$?

Recently I've been writing integrals in the following way, for example $$\int\limits_{[0,1]} {{t^{y - 1}}{{\left( {1 - t} \right)}^{x - 1}}dt} $$ instead of $$\int\limits_0^1 {{t^{y - 1}}{{\left( ...
4
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1answer
326 views

The definition of the logarithm.

One usually gets several definitions of the logarithm along his studies. You might be first introduced to the exponential and then told that the logarithm is its inverse. You might be given $$\log ...
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1answer
262 views

Is how I treat $dx^2$ how you do it?

Sometimes I come up with an integral with a $dx^2$ term. Whenever I have this, I omit the integral with the $dx^2$ with the idea that it's negligible relative to other integrals with only a $dx$ ...
4
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3answers
54 views

Integral with a dependent range

For some constant $a \in (0,1)$, Wolfram-Alpha yields $$\int_{a^2} ^\sqrt a\frac{a}{x} \text{dx} =-\frac{3}{2} a \log(a)$$ How does one approach such an integral? I feel like the solution is ...
4
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2answers
10k views

Why is $dy dx = r dr d \theta$ [duplicate]

Possible Duplicate: Explain $\iint \mathrm dx\mathrm dy = \iint r \mathrm d\alpha\mathrm dr$ I'm reading the proof of Gaussian integration. When we change to polar coordinates, why do we ...
4
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1answer
283 views

Existence of an Analytic Formula for a Definite Integral

It would be very helpful if the following definite integral or a similar one had an analytic solution: $$\int_{-\infty}^{\infty}\mathrm{sech}^2(x)\exp(-\alpha x^2)\,\mathrm dx,\qquad \alpha \geq 0$$ ...
4
votes
1answer
716 views

Dyson series and T product

One of the most important tool in quantum mechanics is the Dyson series because it is the basis of the perturbative theory. There is a step in the derivation that I can't understand. $\{H(t_i)\}$ are ...
4
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1answer
542 views

Question on the Cauchy principal value integral

Motivated by this wiki page, I put my question here: How to prove $$\lim_{\varepsilon\rightarrow 0^+} \int_a^b \frac{x^2}{x^2+\varepsilon^2} \, \frac{f(x)}{x}dx=p.v.\int_a^b \frac{f(x)}{x}dx$$ where ...
4
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1answer
197 views

Power Mean Random Distribution

I'm trying to find a the distribution for the power mean of $n$ random variables on $[0,1]$. I've got the arithmetic mean: $\frac{n}{(n-1)!}\sum_{k=0}^{\lfloor ...
4
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2answers
312 views

Substitution Rule for Definite Integrals

I'm working on an integration by parts problem, and I'm trying to substitute to simplify the equation: $$\int_\sqrt{\frac{\pi}{2}}^\sqrt{\pi} \theta^3 \cos(\theta^2) d\theta$$ Using the substitution ...
4
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1answer
33 views

Measure Theory - Problem with definition about simple functions

I did this question lately and then realized what my mistake was. I got a good help! But looks like only now I understand what was the real problem, to begin with. And this brought me to the same ...
4
votes
1answer
54 views

Show three ways that $f(z)=\frac{\overline{z}}{z-1}$ is not analytic

I need to show the complex function $$f(z)=\frac{\overline{z}}{z-1}$$ is not analytic in three ways; using Cauchy's equations, geometrically, and by integrating over the circle of radius 2. I used ...
4
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1answer
62 views

Approximating a piecewise continuous function with a function in $\mathcal{C}^{\infty}_{0}(\mathbb{R})$

Let $\eta \in \mathcal{C}^{\infty}_{0}(\mathbb{R})$, where $\mathcal{C}^{\infty}_{0}(\mathbb{R})$ is the set of compactly supported infinitely differentiable function, be a function which is ...
4
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1answer
38 views

power series for $\int_0^x e^{-t^2}dt$

Use a known power series expansion to find the power series representation of the integral function $g(x) =\int_0^x e^{-t^2}dt$ centered at $a=0$ My approach Note that $g'(x) = e^{-x^2}$. ...
4
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1answer
47 views

When is an oscillating integral small?

I hope, the title is not too confusing. My question is the following: We all know the Riemann-Lebesgue-Lemma stating that for $f\in L^1(\mathbb R)$, one has $$ \lim_{k\to\infty} \int ...
4
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1answer
214 views

Evaluating $\int \arccos\bigl(\frac{\cos(x)}{r}\bigr) \, \mathrm{d}x$

The title says it all, really - I am looking for $$\int \arccos\left(\frac{\cos(x)}{r}\right) \, \mathrm{d}x$$ where $0<r<1$ and $x$ is in a domain where the integrand is real. It came up ...
4
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1answer
56 views

How to integrate $\int \frac{1}{\sqrt{1+29x^2+100x^4}}dx$ and $\int \frac{1}{\sqrt{1-2x^2-8x^4}}dx$ using elliptic functions?

How to integrate $$\int \frac{1}{\sqrt{1+29x^2+100x^4}}dx$$ and $$\int\frac{1}{\sqrt{1-2x^2-8x^4}}dx$$ using elliptic functions? I have tried to use them, but I got incorrect formula ...
4
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1answer
72 views

Triple integral of $|z|$

I have to calculate the $$\int_A |z| \,dx dy dz $$ with $A=\{(x,y,z): x^2+y^2+z^2\le4, x^2+y^2-2y\le0\}$. Do I use cylindrical or spherical coordinates?
4
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1answer
46 views

Odd Function Integral?

I know if $f$ is an odd function then $$\int_{-L}^L f(x)\:dx = 0$$ my question is, is the converse necessarily true? Intuitively, I feel it should be that by assuming that the integral with those ...
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2answers
95 views

How to do integration of this?

$$\int_0^\infty\frac{x \sin x }{(x^2 + a^2)(x^2 + b^2)}dx\quad\quad a > b > 0$$ I have no idea how to compute this. Any help would be greatly appreciated.
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1answer
39 views

Analysis: the number of sign changes of an integrable function

Let $a,b \in \mathbb{R}$ with $a < b$ and let $f : [a,b] \rightarrow \mathbb{R}$ be a continuous function which is not identically 0. Suppose for some $n \in \mathbb{N}$ that for all $k \in \{0, ...
4
votes
2answers
85 views

how to compute this definite integral if possible?

how to solve this integral? $$\int_0^a\int_0^a\frac{dx\,dy}{(x^2+y^2+a^2)^\frac{3}{2}}$$ my attempt $$ \int_0^a\int_0^a\frac{dx \, dy}{(x^2+y^2+a^2)^\frac{3}{2}}= ...
4
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3answers
85 views

Integral of $e^{\frac{y}{x}}$

How can we find $\int e^{\frac{y}{x}}dy$. An explanation of the answer would be helpful. The answer I got is $ x e^{y/x}$. But not sure about the steps used for obtaining the answer...
4
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1answer
69 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 ...
4
votes
2answers
97 views

Evaluate $\int\frac{1}{1+x^6} \,dx$

I came across following problem Evaluate $$\int\frac{1}{1+x^6} \,dx$$ When I asked my teacher for hint he said first evaluate $$\int\frac{1}{1+x^4} \,dx$$ I've tried to factorize $1+x^6$ ...
4
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2answers
76 views

Asymptotic expansion for Fresnel Integrals

If you take the fresnel integrals to be $$S(x) = \int_{0}^{x}\sin \left(\frac { \pi \cdot t^2}{2} \right) dt$$ How do you find the asymptotic expansion? I know it begins with a $1/2$ but how?
4
votes
1answer
56 views

Is there an alternative way to solve this integral?

I was given the integral $$\int \frac{2}{e^{-x}+1}dx$$ Here is my method to get the (correct) solution: $$\int \frac{2}{e^{-x}+1}dx$$ $$=2\int \frac{1}{e^{-x}+1}dx$$ $$=2\int ...
4
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1answer
43 views

Why can I not combine integrals this way?

Evaluating the triple integral $\int^1_0 \int^{1-z}_0 \int^{1-y-z}_0 \text{dxdydz}$, I get $\frac 16$. Evaluating the triple integral $\int^1_0 \int^1_0 \int^1_0 \text{dxdydz}$, I get $1$. So I ...
4
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2answers
83 views

How to calculate the value of the special integral

I get $${\left. \frac{\partial ^2}{\partial n^2} \left( \frac{\partial ^2}{\partial m^2} B(m,n) \right) \right|_{m = \frac{1}{2},n = 0}} = \int_0^1 \frac{\ln^2 x \ln^2 (1 - x)}{\sqrt x (1 - x)} \, dx ...
4
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1answer
188 views

Books with collections of unusual and advanced integration techniques

I am searching for some comprehensive books that collect, explain, and provide examples of extremely advanced and/or unusual integration techniques. Can you point out some good references? Note: ...
4
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1answer
77 views

continuous-time version of fatou's lemma

I have just read a textbook on stochastic processes that implicitly uses the fact that \begin{equation} \int \liminf_{t \to \infty} f_t \leq \liminf_{t \to \infty} \int f_t, \end{equation} for ...
4
votes
2answers
98 views

What is wrong with the following u-substitution?

We will calculate $\displaystyle\int^{2 \pi}_0 x \, dx$. Let $u=\sin (x)$, and observe that $\sin(2 \pi)=0$ and $\sin(0)=0$. We also have that $\frac{du}{dx}=\cos(x)=\sqrt{1-u^2}$. Hence, $$ \int^{2 ...
4
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1answer
105 views

Area and volume relation (multivariable calculus problem)

Let $D \subset R^3$ a region over the plane $z=0$, if $C$ is the cone of base $D$ and vertex at $(0,0,1)$, show that $Vol(C)=\dfrac{1}{3}A(D)$, where $A(D)$ is the area of the region $D$. First I ...
4
votes
2answers
80 views

Computing the integral $\int \frac{u}{b - au - u^2}\mathrm{d}u$

After working on an ODE I find I am needing to solve the integral $$\int \frac{u}{b - au - u^2}\mathrm{d}u$$ Trig subs, banging heads against walls, and sobbing have not yielded a solution. Yet. ...
4
votes
1answer
88 views

Evaluate $\int_2^4\frac{\sqrt{x^2-4}}{x^2}\mathrm dx$

Evaluate $$\int\limits_2^4\frac{\sqrt{x^2-4}}{x^2}\mathrm dx$$ My working: $x=2\sec\theta\quad\Rightarrow\quad\theta=\arccos\left(\frac{2}{x}\right)$ $dx=2\sec\theta\tan\theta d\theta$ ...
4
votes
4answers
215 views

Definite integral $\int_{0}^{\infty}e^{-u}\frac{1}{\left(\sqrt{1+(h+u)^{2}}\right)^{5}}du$

Hi guys I have the following definite integral to solve: $$\int_{0}^{\infty}e^{-u}\frac{1}{\left(\sqrt{1+(h+u)^{2}}\right)^{5}}du$$ is it possible to obtain an analytic expression? And if not why? ...
4
votes
1answer
49 views

Double Integral transformation to Polar coordinates

Here's the question from an exam that I couldn't solve: If $\int_1^2 \int_0^x \frac{1}{(x^2+y^2)^\frac{3}{2}} ~\mathrm{dy} ~\mathrm{dx}$ transforms to $\int_0^a \int_b^c \frac{1}{r^2} ...
4
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2answers
65 views

p-norm of a function

Let $f\in L^1(\mu)\cap L^\infty(\mu)$. I have proved for any $1<p<\infty$, $f\in L^p(\mu)$, $w(p)=||f||_p$ is continuous w.r.t. $p$, and $\lim_{p\to \infty}||f||_p=||f||_\infty$. Is $w(p)$ ...
4
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1answer
39 views

an argument that strengthen Lusin's theorem

Let $f$ be a measurable function on a subset $E$ of $\mathbb{R}^n$. Lusin's theorem states that for any $\epsilon>0$, there exists a measurable subset $F$ such that $F$ open in $E$, ...
4
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1answer
147 views

Double integral containing $e^{(b+ic)/z^2}$

I want to solve the two integrals \begin{aligned} I_3\,& = \int_{0}^{\infty} ze^{a/z^2 - z^2} dz\\ I_4\,& = \int_{0}^{\infty} \frac{1}{z}e^{a/z^2 - z^2} dz. \end{aligned} where ...
4
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1answer
161 views

Multiple integral over a disc

I would need some help on this integration problem: $$I=\int_0^{2\pi}\int_0^{R}\int_0^{2\pi}\int_0^{R}\exp(-a\ r_{12}) \ r_1 \ r_2 ...
4
votes
1answer
91 views

Double integral involving zeta function: $\int_0^\infty \frac{1-12y^2}{(1+4y^2)^3}\int_{1/2}^{\infty}\log|\zeta(x+iy)|~dx ~dy.$

I'm having trouble evaluating the following double integral: $$\int_0^\infty \frac{1-12y^2}{(1+4y^2)^3}\int_{1/2}^{\infty}\log|\zeta(x+iy)|~dx ~dy.$$ Do please remark that $\zeta$ is the zeta ...
4
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1answer
120 views

Problematic integral $\int_0^\pi \frac{x\sin x}{1+\cos^2x}\ dx$

How to calculate $$\int_0^\pi \frac{x\sin x}{1+\cos^2x}\ dx\ ?$$ I wish I could say I ran out of ideas, but actually I have none.
4
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2answers
75 views

2 calculus questions with integration - check me

I have 2 questions I would like assistance with. 1) Find the area of the region bounded by the graphs $y=5x, y=15x, y=\frac{4}{x}, y=\frac{8}{x}$ This was very difficult and tedious. I had trouble ...
4
votes
1answer
75 views

Find general solution of first order non-linear in a transcendental function

I have the function $$\frac{dV}{dT}=1-V^2$$ Just looking to see if my working is okay. $$dV=1-V^2dT$$ $$\frac{1}{1-V^2}dV=dT$$ Integrate $$\int{}\frac{1}{1-V^2}dV=\int{}dT$$ Let $V=\tanh(x)$ ...
4
votes
1answer
55 views

Tricky looking integration (after separation of variables)?

I've come across something in my notes that jumps from: $${d\rho \over dz} = \sqrt{\left({\rho \over C}\right)^2 - 1}$$ to: $$\rho(z) = C \cosh\frac{z-z_0}{C}$$ I know that separation of variables ...
4
votes
2answers
104 views

On the convexity of a particular form of integrals

EDIT: I made some critical corrections below. Let $\mathcal{H}\colon\mathbf{w}\cdot\mathbf{x}+c=0$ be a hypeplane in $\mathbb{R}^n$. Also, let $g\colon\mathbb{R}^n\to\mathbb{R}_+$, be a non-negative, ...
4
votes
2answers
93 views

Taylor series of an integral function

Problem $$I(x) = \int_{1}^x \frac{e^t - 1}{t}$$ Find $I'( \sqrt{x} )$. Solution We know that $F'(x) = f(x)$ by the fundamental theorem of calculus so $$I'(x) = \frac{e^t -1}{t}$$ And so $$I'( ...