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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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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66 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
115 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 ...
0
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2answers
33 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.
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1answer
38 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
47 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
36 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
253 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
68 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
32 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
32 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
50 views

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
50 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 ...
12
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1answer
142 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
50 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
84 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
67 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
76 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 ...
1
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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
142 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
votes
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
votes
3answers
70 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
31 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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0answers
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
64 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
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3answers
127 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
94 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
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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
21 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?
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54 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 ...
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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) = ...
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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 ...
1
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0answers
20 views

Will Gauss quadrature numerical integration work with a variable dx

The question kind of says it all, but I'm reading about Gauss quadrature from here: http://www.damtp.cam.ac.uk/lab/people/sd/lectures/nummeth98/integration.htm which gives an equation of this form: ...
2
votes
3answers
103 views

What is $\operatorname{Ei}(x)$?

I was trying to solve $$\int\frac{e^x - e^{-x}}{x}\,dx$$ But I have no idea how to do it and the calculator said to use a common integral that I don't know what it means.
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2answers
207 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 ...
2
votes
1answer
44 views

Positive Linear Transformations: What good for?

Positivity is a concept appearing quite frequently in the study of algebras and its related spectral theory. Positive elements naturally give rise to an ordering and therefore allows to construct ...
0
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1answer
43 views

Integrating probabilities

My following problem is of general nature, here is an example to illustrate it. For example let $\left(\xi_i\right)_{i \geq 1}$ be independent and identically Exp(1) distributed random variables. We ...
1
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2answers
53 views

Find the volume below $\sqrt{x}+\sqrt{y}+\sqrt{z}=1$ in the first quadrant

I understand that we have to use transformation $$x = u^2, y = v^2, z = w^2$$ but I cannot figure out the limits. I just need a rough sketch of how to approach this. Could anyone give me some ideas?
2
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1answer
82 views

Antiderivative of $\exp(x^2)$ [duplicate]

Can you please provide a step by step solution for next integral. I don''t have any idea of hoe this can be solved $\displaystyle\int e^{x^2}\,dx$.
13
votes
2answers
91 views

Integrals of integer powers of dilogarithm function

I'm interested in evaluating integrals of positive integer powers of the dilogarithm function. I'd like to see the general case tackled if possible, or barring that then as many particular cases as ...
4
votes
2answers
62 views

Integral of the Square of the Elliptic Integral

Someone must know a good technique for $$ \int E^{2}(x)dx $$ Where $E$ is the complete elliptic integral of the second kind: $$ ...