All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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1answer
30 views

Bochner: Absolute Integrability

For a Bochner measurable function it holds: $$f\text{ Bochner integrable}\iff\|f\|\text{ Bochner integrable}$$ for any positive measure $\lambda\geq 0$. The one direction is relatively simple when ...
1
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0answers
9 views

Integral of inverse of square root of quartic function with real roots

I was doing a physics problem and in order to finish it, I need to prove that: $\int_{x1}^{x2}\frac{dx}{{((x - x1)(x - x2)(x - x3)(x - x4))}^{1/2}} = \int_{x3}^{x4}\frac{dx}{{((x - x1)(x - x2)(x - ...
2
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1answer
26 views

integral calculate by complex analysis methods

Calculate using methods from comples analysis. $$ \int_0^{2\pi} \,\sin ^{2n} \phi\, d\phi$$ So this is how I started: $$\sin^{2n} \phi = \left[\frac{e^{i \phi}-e^{-i \phi}}{2i}\right]^{2n} = ...
0
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0answers
59 views

Solved! Definite Integration and Area. I can't reach the given solution.

The problem I encounter is to reach the same solution I've been given! The function is: $f(x) = \left \{ \begin{matrix} x^2 & \mbox{}x\mbox{ < 1 } \\ \dfrac{1}{x} & ...
3
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4answers
72 views

Prove that $\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}$

Prove that \begin{equation} \int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24} \end{equation} I tried to use by parts method and ended with \begin{equation} \int \ln^2(\cos ...
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0answers
17 views

Why is $\int_0^{2\pi} e^{i\,k\rho[\sin\alpha\cos\alpha-\sin\theta\cos(\phi-\beta)]}\mathrm{d}\beta = 2\pi J_0(k\rho\xi)$?

The following is an integral in Jackson Classical Electrodynamics (3rd ed.). In equation (10.112) the integral $$ \int_0^{2\pi} ...
1
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1answer
44 views

Spectral density of a sample covariance matrix in a Gaussian Random Ensemble

Let $N > 0$ and $T > N$ be integers and $C$ be a real, symmetric $N \times N$ matrix.The question is to compute the following integral: \begin{equation} U_{N,T}(t) := \frac{1}{N} ...
3
votes
1answer
73 views

What does this paper mean by “$f(x)$ is practically a rational function”?

The paper "Infinite integrals involving Bessel functions by contour integration" by Qiong-Gui Lin gives a method to solve integrals of the form $\intop_{0}^{\infty}x^{v}f(x)J_{v}(qx)\, dx$. One of the ...
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0answers
18 views

integration formula

help me please all these functions are regular. How we can found this formulation $$ \displaystyle\int_{\Omega} (f(u)-f(k)) \nabla p(g(u)-g(k)) \xi dx = - \displaystyle\int_{\Omega} H(u,k) \nabla \xi ...
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2answers
218 views

Integral $\int_0^{\Large\frac{\pi}{4}}\left(\frac{1}{\log(\tan(x))}+\frac{1}{1-\tan(x)}\right)dx$

I am wondering if anyone would know how to evaluate this integral: $$\int_{0}^{\Large\frac{\pi}{4}}\left(\frac{1}{\log(\tan(x))}+\frac{1}{1-\tan(x)}\right)dx.$$ I've tried, unsuccessfully, the change ...
13
votes
1answer
169 views

Prove $\displaystyle \int_{0}^{\pi/2} \ln \left(x^{2} + \ln^{2}\cos x \right) \, dx=\pi\ln\ln2 $

How to prove\begin{equation} \int_{0}^{\pi/2} \ln \left(x^{2} + \ln^{2}\cos x \right) \, dx=\pi\ln\ln2 \end{equation} I don't know how to answer it. When I asked this integral to my brother, ...
5
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0answers
49 views

How prove $f(a_{i})=0$ if $\int_{0}^{1}x^kf(x)dx=0,k=1,2,3,\cdots,n$ [duplicate]

let $f(x)$ is continuous on $[a,b]$,and such $$\int_{0}^{1}x^kf(x)dx=0,k=0,1,2,3,\cdots,n$$ show that: there exsit $n+1$ different $a_{1},a_{2},\cdots,a_{n},a_{n+1}(a_{i}\neq a_{j},\forall ...
5
votes
1answer
63 views

What is an example of a function that is measurable but not strongly measurable?

Let $(\Omega, \Sigma)$ be a measurable space and $X$ a Banach space. Let $f: \Omega \rightarrow X$. $f$ is called measurable if every the preimage of every Borel set in $X$ is an element of ...
2
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2answers
83 views

Solving Bessel integration

What would be the solution of the bessels equation, $$b=k A(t)\int_0^{\infty} J_0 (k \rho) e^ \frac{-\rho^2}{R^2} \rho d \rho$$ Can I sove that by using this formulation? $$c= \int_0^{\infty}j_0(t) ...
0
votes
1answer
47 views

How to show $\int\frac{d}{dx}(a^u)dx=a^u+C$ more rigorously?

We all know that $$\int\frac{d}{dx}(a^u)dx=a^u+C$$ where I am differentiating with respect to $x$. But how can I write it in a more rigorous way like for example using the Fundamental Theorem of ...
1
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0answers
17 views

Expressing indefinite integrals in terms of a predefined set of functions.

It is well known that some integrals of elementary functions cannot be expressed as elementary functions. I was wondering if it was possible to extend the set of elementary operators by some ...
4
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2answers
70 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. ...
2
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1answer
26 views
+50

Calculating $\text{D}g$ of $g(x,y) = \int_\frac1x^1\frac1t\exp(t^3x^2y)\text{d}t$

Let $g:(1,\infty)^2\to\mathbb{R}$ be given by $$g(x,y) = \int_\frac1x^1\frac1t\exp(t^3x^2y)\text{d}t.$$ How can I calculate $\text{D}g$ using parameter-dependent integrals?
3
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2answers
62 views

And another real integral to be solved by contour integration

I want to solve $$\int_0^\infty\frac{1}{x^3+x^2+x+1}dx$$ and i have really learned a lot already by failing to solve it. I want to solve it using a clever contour. It is possible to do it using ...
2
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1answer
177 views

Integration of this type?

Can anyone help me with this? I used completing the square but do not know how to continue? Thanks 9.. Gaussian Integral The following definite integration is particularly relevant in the ...
6
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2answers
541 views

Is there a change of variables formula for a measure theoretic integral that does not use the Lebesgue measure

Is there a generic change of variables formula for a measure theoretic integral that does not use the Lebesgue measure? Specifically, most references that I can find give a change of variables ...
3
votes
3answers
255 views

Evaluating $\int \frac{1}{x\sqrt{9x^2-1}}\,dx$

I try to integrate $$\int \frac{1}{x\sqrt{9x^2-1}}\,dx$$ let $u=x^2,\quad \quad du=2x\,dx,\:\quad \:dx=\frac{1}{2x}\,du$ $$ \begin{align} & \int \frac{1}{x\sqrt{9u-1}}\frac{1}{2x}\,du \\[8pt] = ...
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votes
0answers
76 views

How can I find $\int_0^\pi\sqrt{5-4\cos(t)}\ dt$? [on hold]

I'm after the value of $$\int_0^\pi\sqrt{5-4\cos(t)}\ dt.$$ Please help me in solving this question. I am not able to solve it.
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votes
2answers
36 views

Issue with relatively simple integral

I'm having trouble with this integral: $$\int_{-\infty}^{\infty} dx/\sqrt(a^2+x^2)^3$$, $a=const$. I know it looks simple, but I've tried $a^2+x^2=t$, and $\sqrt(a^2+x^2)=t$, and those didn't work.
2
votes
1answer
41 views

What exactly is integration?

Consider the function $y=2x$. The graph of this function is here. Next, Consider $\int 2x dx=x^2 + c$. Here is the graph : http://www.wolframalpha.com/input/?i=plot+y%3Dx^2+from+-2+to+2. My ...
1
vote
1answer
51 views

Evaluate line integral without parameterizarion

It's been brought to my attention that line/surface integrals and integrals of differential forms in general can be evaluated without introducing a parameterization, however I haven't been able to ...
1
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2answers
38 views

Integration of function (inverse)

Does anyone know how do I start on part (b)? Thanks
1
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1answer
38 views

Integration of this funtion

No matter how hard I try, I cannot get to prove it. I understand we have to use the factor formula of trigo. But still cant prove it. Please help, thanks
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2answers
33 views

Constructing a polynomial bump function

Proposition: Suppose $f$ is continuous and $\int_a^bf(x)x^ndx = 0$ for all $n$. Then $f$ is zero on $[a,b]$. This can be proven by uniformly approximating $f$ with polynomials via the Weierstrass ...
0
votes
1answer
20 views

Laplace transform quick answer check :) using second shift theorem

I want to get $L((t-4)^2u(t-4))$ I say this is a second shift with $g(t)=(t^2-4t)$ and my friend says "NO you are wrong, you are dumb!!!!!! $g(t)$ is MOST CERTAINLY equal to $t^2$" Mine gives me ...
6
votes
1answer
95 views

A closed-form of $\frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx$

I am looking for a closed-form of this integral \begin{equation} \frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx \end{equation} I ...
4
votes
2answers
88 views

Trying to Integrate$ \iint xy\log|x-y|\, dy\,dx $

Hello I am trying to integrate $$ I:=\int_{a}^{b}\int_{a}^{b}xy\log\left(\,\left\vert\,x - y\,\right\vert\,\right) \,{\rm d}y\,{\rm d}x,\qquad 0 < a <b $$ for $x,y\in \mathbb{R}$. I added the ...
1
vote
1answer
89 views

Bessel function and upper bound

I'm stuck on this following problem: Let $G$ a function such that $0\leq G(t)\leq 1$, and $G(t)=1$ if $B^2\leq t\leq 4B^2$, with $\operatorname{supp}G\subset [\frac{1}{4}B^2, 9B^2]$ and $G^{(j)}\ll ...
1
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1answer
341 views

Bessel integral solution or simplification

I am trying to verify the following formula involving Bessel functions of the first kind and am having no luck. The formula is $$ \int{\omega} J_n(\rho \omega)\mathrm d\omega = \frac {1} {\rho} ...
3
votes
2answers
102 views

Is This a Bessel Function?

Is the function $$y(x) = c \int_{-1}^1 \cos(xt)(1-t^2)^{n-\tfrac{1}{2}}dt = c \sum \tfrac{(-1)^m x^{2m}}{(2m)!} \int_{-1}^1 t^{2m}(1-t^2)^{n-\tfrac{1}{2}}dt$$ given here a bessel function? It ...
10
votes
3answers
273 views

$\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}.$

Hi I am trying to prove the relation $$ I:=\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}. $$ I tried expanding the log argument by using $\sin x/ \cos x=\tan x,$ and than used ...
2
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0answers
93 views

Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration $$ \int_{0}^{1}{\ln\left(\,x\,\right)\over \,\sqrt{\vphantom{\large A}\,1 - x^{2}\,}}\,{\rm d}x $$ I am not sure as to how to work with the branch ...
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votes
3answers
68 views

How to evaluate an integral of the form $\int \frac{dx}{-ax^2 + b}$?

I need to evaluate $\int \frac{dx}{-ax^2 + b}$ while both $a$ and $b$ are positive. I was blocked while I was trying $ x=\tan\theta $ which turned $ dx=\sec^2\theta\, d\theta $ This method didn't ...
3
votes
1answer
33 views

Evaluate the integral $\iiint\limits_E x^2 \,\, \mathrm{d}V$

Where E is the region bounded by the xz-plane and the hemispheres $y=\sqrt{9-x^2-z^2}$ and $y=\sqrt{16-x^2-z^2}$. This is an exercise from my professor guide. What I tried so far: These exercise ...
2
votes
2answers
139 views
+50

Verifying an antiderivative found in any integral table

If $a > 0$, and $0 < b < c$. \begin{equation*} \int \frac{1}{b + c\sin(ax)} \, {\mathit dx} = \frac{-1}{a\sqrt{c^{2} - b^{2}}} \, \ln\left\vert\frac{c + b\sin(ax) + \sqrt{c^{2} - ...
1
vote
0answers
22 views

Weak stochastic integral

I recently encountered the following object, referred to as "weak stochastic integral" in the book of SPDE's by Prévôt/Röckner [PR07]: $$ \int_0^T \langle \Psi \,\mathrm dW(t), \Phi(t)\rangle $$ A ...
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votes
1answer
29 views

Error for Simpson's 3/8 Integration Rule [on hold]

I couldn't find the derivation for error of simpson's 3/8 Integration Rule. Can anybody help me derive it please?
5
votes
1answer
51 views

Convergence of Integral near 0

I am trying to determine the convergence of the integral \begin{equation} \int_0^1 \frac{f(x)}{x}\, dx \end{equation} given that $f(x)$ is bounded and continuous on $[0,1]$, and that $f(x)=0$. The ...
0
votes
1answer
117 views

Proof of Gauss' Law of gravitation without reference to Newton?

Gauss' Law of gravity is: $$\bigtriangledown \cdot \mathbf{g}= 4\pi G\rho$$ This can be shown to be equivalent to Newton's Law of gravity via the divergence theorem. However, this does not really ...
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1answer
3k views

Volume of a Cylinder Using Cylindrical Coordinates and Triple Integration

Calculate the Volume $V$ of a right circular cylinder of radius $a$ and height $h$, using cylindrical coordinates and triple integration.
0
votes
3answers
116 views

How to evaluate $\int{d(y^2)}$?

Can anybody help me to solve this integral please: $$\int{dy^2}$$ Here $dy^2$ means $d(y^2)$, not $(dy)^2$. Thanks for any help.
1
vote
2answers
37 views

Integrable combinations - I can't seem to arrive at the given answer

I need help! I can't seem to arrive at the answer given in our textbook. I'm new here, so I really need help. The instruction says that I need to solve this D.E by recognizing integrable ...
37
votes
5answers
891 views
+200

How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$ I suspect it might exist because there are similar integrals having closed forms: ...
7
votes
1answer
105 views

Quaternion integration

If the angular velocity is changing continuously, the following holds true $ q(t)=q(0)\exp\left({\int_{0}^{t}\frac{q_\omega(\tau)}{2}\ d\tau}\right) \tag 1$ Specifications and Data $q(t),q(0)$ ...
2
votes
1answer
144 views

How do I Solve This Kind of Differential Equation? [on hold]

How do I solve this differential equation? $$y(2x+y^2)dx+x(y^2-x)dy=0$$