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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Integration by Parts: When can you not use the Table Method. Why?

I'm learning about integration by parts, primarily from Stewart's text (7th edition). In a supplemental book I have it brings up something called the Table Method. I really find this method appealing ...
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0answers
14 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 - ...
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18 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} ...
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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} = ...
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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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4answers
73 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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2answers
32 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 ...
5
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0answers
50 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 ...
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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 ...
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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 ...
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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.
2
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1answer
181 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 ...
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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
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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 ...
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 ...
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
3
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3answers
256 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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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 ...
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. ...
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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 ...
3
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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 ...
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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?
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0answers
60 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} & ...
1
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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 ...
13
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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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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 ...
4
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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 ...
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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.
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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 ...
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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 ...
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2answers
82 views

Trigonometric integral: $\int_{25\pi/4}^{53\pi/4} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx$

Is it possible to evaluate the following in a closed form? $$\int_{25\pi/4}^{53\pi/4} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx$$ I found the above definite integral at I&S but the solution is ...
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1answer
17 views

Laplace transform convolution attempt

I can't seem to get this Laplace working using the convolution method. $H(s) = \frac{1}{s^2(s+2)}$ Which I can't get to work using convolution. So I am separating it into $\frac{1}{s^2} * ...
2
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1answer
144 views

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

How do I solve this differential equation? $$y(2x+y^2)dx+x(y^2-x)dy=0$$
2
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1answer
38 views

Calc II - Definite integral of sqrt(t^2 + t) from 2x to 1?

How do I find $$\int_1^{2x}\sqrt{t^2 + t}$$ with only knowledge from a Calculus I course? I've tried plugging this puppy into Wolfram Alpha and other integral solvers, which report it as solvable ...
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1answer
25 views

When can we exchange sum limit and integral

A simple question: when can wo exchange sum and integral? $$\sum_{n=0}^\infty\int f_n(x)dx=\int\sum_{n=0}^\infty f_n(x)dx=\int f(x)dx$$ $$\lim_{n\rightarrow \infty}\int f_n(x)dx=\int\lim_{n\rightarrow ...
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1answer
20 views

Laplace transform on a non-standard sort of problem

I don't know where a laplace comes into play here: $\ddot{a}+2a=0,a(0)=b_1,\dot{a}(0)=b_2$ I am meant to solve the above using a Laplace transform, but I don't see how I would use it here? I ...
3
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2answers
42 views

How to simplify the integral of $\int\frac{\cos(8x)}{\cos(4x)+\sin(4x)}dx$?

So I am trying to integrate this problem $\int\frac{\cos(8x)}{\cos(4x)+\sin(4x)}dx$, and my professor went over it in class and went from $\int\frac{\cos(8x)}{\cos(4x)+\sin(4x)}dx \rightarrow ...
3
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3answers
74 views

Integral of $\cos(\cos x)$ over $[0,2\pi]$

How to compute the following integral? $$\mathcal{J}_2=\int_{0}^{2\pi}\cos(\cos t)\,dt$$ I'm trying to compute this integral, but I have no idea of how to do it, can someone help me?
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0answers
34 views

Help with functional integral

I'm stuck on how to do a functional integral. The integration I'm trying to do is of this form $\frac{\partial}{\partial B(\tau)} \left[ \exp\left(-B^2(\tau)\right)+\int_{0}^{\tau} ...
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24 views

Integrating differential forms over a box

I've only ever seen examples of integrating a differential form over a curve C involving defining a parameterization. I have seen people integrate 1 forms over a box without defining a ...
6
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2answers
65 views

Trigonometric functions expressed as definite integrals with Bessel functions

Prove that $$\frac{\sin(x)}{x}=\int_0^\frac{\pi}{2}J_0(x\cos(\theta))\cos(\theta)\,d\theta \tag{a}$$ $$\frac{1-\cos(x)}{x}=\int_0^\frac{\pi}{2}J_1(x\cos(\theta))\,d\theta \tag{b}$$ Hint: ...
2
votes
3answers
129 views

Limit of a Riemann Sum and Integral

I've been trying to solve this problem, but I haven't been able to calculate the exact limit, I've just been able to find some boundaries. I hope you guys can help me with it. Let $f:[0,1] \to ...
6
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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 ...
1
vote
1answer
49 views

Find the coefficients in quadrature formula on $[0,1]$ with the nodes at $1/4$, $1/2$, $3/4$

In my worksheet I was given a question about numerical integration that says: Find the formula for $\int_{0}^{1}f(x)dx=A_{0}f(\frac{1}{4})+A_1f(\frac{1}{2})+A_2f(\frac{3}{4})$ I suppose the goal ...
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0answers
44 views

How can I calculate the force that is applied on a tube by an another tube?

Let's say there is two tubes (cylinders with no tops or bottoms) with charges $q_1$ and $q_2$, radii $b_1$ and $b_2$, lengths $\ell_1$ and $\ell_2$. These tubes are located along the axis of each ...
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?
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 ...
6
votes
3answers
138 views

Find $\int_0^\pi \sin(x)\,dx$ explicitly

A book asks me to prove that: $$\int_0^{\pi}\sin(x)\,dx = 2$$ Using the identity: $$\sin\left(\frac{\pi}{n}\right) + \sin\left(\frac{2\pi}{n}\right) + \cdots + \sin\left(\frac{n\pi}{n}\right) = ...
4
votes
0answers
72 views

is there closed form for $\int_0^{\pi/4}\exp(-\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a})dx$

Is there closed form for $$I(a)=\int_0^{\pi/4}\exp(-\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a})dx $$where is $a\in (-1,3)$ I've tried with $\tan x=u$ and I got the result of sum in term of ...