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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invariance of integrals for homotopy equivalent spaces

I just wanted to know whether the integral of a closed n-form is invariant if we integrate it over homotopy equivalent spaces. This seems like a generalization of "Homotopy invariance of line integral ...
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2answers
37 views

How to find the antiderivative of this function?

I want to integrate this function. . $$\int \dfrac{x^2}{e^x-1} \, dx $$ I used integration by parts formula to integrate it. However I have reached somewhere where I got something like this : $$\int ...
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3answers
51 views

Integrating $x^3\sqrt{ x^2+4 }$

Trying to integrate $\int x^3 \sqrt{x^2+4 }dx$, I did the following $u = \sqrt{x^2+4 }$ , $du = \dfrac{x}{\sqrt{x^2+4}} dx$ $dv=x^3$ , $v=\frac{1}{4} x^4$ $\int udv=uv- \int vdu$ $= ...
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1answer
45 views

How find this integral $I=\int_{-\pi}^{\pi}\frac{x\cdot \sin{x}\cdot arccot{(2014^x)}}{1+(\cos{x})^4}dx$

Question: Find this integral $$I=\int_{-\pi}^{\pi}\dfrac{x\cdot \sin{x}\cdot arccot{(2014^x)}}{1+(\cos{x})^4}dx$$ let $x\to -x$,so $$I=\int_{-\pi}^{\pi}\dfrac{x\sin{x} \cdot ...
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1answer
14 views

Find the Laplace transform of integral(from 0 to x) sin(2t) dt

Find the Laplace transform of $\int_0^x\,\sin\,(2t)\,dt$ So basically, $$\int_0^x\,\sin\,(2t)\,dt = -\frac{1}{2}(\cos\,(2x) - 1)$$ So $$\mathcal{L}\{\cos\,(2x)\} = \dfrac{s}{s^2 + 4}$$ ...
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5answers
239 views

Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
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3answers
35 views

Calculus 2 Integral of$ \frac{1}{\sqrt{x+1} +\sqrt x}$

How would you find the integral of $1/(\sqrt{x+1} + \sqrt x) dx$. I used u substitution and got this far: $u = \sqrt{x+1}$ which means $(u^2)-1 = x$ $du = 1/(2\sqrt{x-1}) dx = 1/2u dx$ which means ...
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2answers
38 views

Integrate $\int\frac{dx}{x\sqrt{x^2+x+1}}$ [on hold]

Hello I need some help with the following integral: $$\int\frac{dx}{x\sqrt{x^2+x+1}}$$ Have been trying u-sub, and parts which do not get me to a solution!
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3answers
37 views

Integrating $\int_{-\pi}^{\pi} \frac{ d\theta}{w - sin \theta}$

I know that the integral $$\int_{-\pi}^{\pi} \frac{ d\theta}{w - sin \theta} = \frac{2\pi}{\sqrt{w^2-1}}$$ where w, is an arbitrary constant and at some point you must do the substitution $$u = tan( ...
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Simple complex line integral over a rectangle

What is the easiest way without using residues to calculate: $$\int_{\gamma} {\overline z \over {8 + z}} dz$$ Where $\gamma$ is the rectangle with vertices $\pm 3 \pm i$ in $\Bbb C$ in the clockwise ...
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2answers
21 views

Spectral Measure: Support

The support of a spectral measure is defined by: $$\mathrm{supp}E:=\bigcap_{C:E(C)=1}C$$ where $C$ are closed subsets (see german wiki). So by definition it is closed. However I'm wondering wether it ...
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1answer
24 views

Calculating area relative to the y-axis

I was asked to calculate the area of the region bounded by the following graph: $$ y = x^2+4x ; y=0$$ I substituted $y$ in order to get $x = 0$ 0r $x=4$. Now I would like a little bit of help to get ...
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1answer
35 views

Spectral Measure Integration: Product

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. Define the integral of simple functions by: $$\int_\Omega ...
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1answer
49 views

Area enclosed by curves

Given the curves $y=x^2$ and $y=\frac{1}{2}(x+x^4)$. What is the area enclosed by them ? I can't find the points of intersection of the curves.
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1answer
97 views

Computing $\lim\limits_{n\to\infty} \Big(\sum\limits_{i = 1}^n \sum\limits_{j = 1}^n \frac1{i^2+j^2}-\frac{\pi}{2} \log(n)\Big)$.

In the chatroom we discussed about the asymptotic of $\displaystyle \sum_{i = 1}^n \sum_{j = 1}^n \frac1{i^2+j^2}$, and if we think of the inverse tangent integral, it's easy to see that ...
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1answer
25 views

Volume generated by lemniscate revolving about a tangent at the pole.

The lemniscates $r^2 = a^2\cos2\theta$ revolves about a tangent at the pole. What is the volume generated by it ? Please explain in detail. I found a couple of answers on finding surface areas, ...
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7answers
99 views

How to integrate $\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}dx$

How can I approach this integral? ($0<a \in \mathbb{R}$ and $n \in \mathbb{N}$) $$\large\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}\, dx$$ Integration by parts doesn't seem to make ...
2
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3answers
283 views

Evaluating the integral $ \int{\frac{x}{\sqrt{2x^2 + 3}}}dx $

I am trying to integrate the following: $$ \int{\frac{x}{\sqrt{2x^2 + 3}}}dx $$ It seems to me to be a trig substitution; however, I couldn't seem to get it into one of the three forms, i.e., ...
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1answer
53 views

Bochner: Lebesgue Obsolete?

Bochner's notion of integral: $$F\text{ Bochner integrable}:\iff \exists S_n\in\mathcal{S}:\quad \int\|S_m-S_n\|\mathrm{d}\mu\to 0\quad(S_n\to F)$$ This version totally circumvents Lebesgue's notion ...
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3answers
35 views

Triple Integral in Spherical Coordinates.

$\newcommand{\de}{\operatorname{d}}$A little stuck on this one. $$\iiint_V ye^{-(x^2+y^2+z^2)^2}\de V$$ Use Spherical Coordinates to evaluate where V is the solid that lies between y=0 and the ...
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2answers
43 views

Elias Stein : Real Analysis

I cannot understand why this particular line in the text is true: " Moreover, there are $O(k^{d-1})$ cubes in $\cal{Q}\ '$ " For the text see ...
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1answer
18 views

Integral on complex plane of a gaussian times power

I can't solve the integral $$ I = \int_\mathbb{R} \int_\mathbb{R} \ (x + i y)^{2k} \ e^{\displaystyle - \frac{(x + i y)^2 R^2}{1+R^2} - y ^2} d x d y $$ which can be rewritten as $$ I= \int_\mathbb{R} ...
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3answers
29 views

Solving using integrating factor [on hold]

Q) Solve $y' = 2x + y$ using the integrating factor. Can anyone guide me with steps here? Help appreciated. Thanks.
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0answers
31 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}x^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ This could be used to evaluate partial sums using knowledge of the ...
3
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1answer
41 views

Calc 2: Integration by Parts w/ trig identities

$$\int e^{3\theta}\sec^4(e^{3\theta})\tan^{11}(e^{3\theta})d\theta$$ I just want to make sure that I'm doing this correctly so that I can understand the material. I would also appreciate any tips or ...
2
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3answers
80 views

Integral $\int_0^\pi \frac{x\,\operatorname dx}{a^2\cos^2x+b^2\sin^2x}$

Integrate: $$ \int_0^\pi \frac{x\,\operatorname dx}{a^2\cos^2x+b^2\sin^2x} $$
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0answers
39 views

Evaluating integral involving product of cosine inverse

I am trying to evaluate the below mentioned integral which involves product of two cosine inverses and two variables $x$ and $y$. I need to evaluate the integral or get an approximate value of this ...
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0answers
12 views

Generalized change of variables in integral

When I read the following (http://www.math.helsinki.fi/~analysis/GraduateSchool/maly/gs.pdf ), it is hard to understand it. In particular, what does it mean by the last equation? Why does it make ...
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5answers
223 views

Why consider square-integrable functions?

Why are $L^2$ functions important? From reading around I have three hypotheses: they show up in QM (but, why?) they form an inner product space (but, is that a "tight bound" or is the class easily ...
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0answers
57 views

Is there a formal proof of this basic integral property?

This has really been bothering me because everywhere I have looked the answer has been "A proof has been omitted because the theorem is very intuitive" or "Proofs are very complicated and not worth ...
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0answers
56 views

Prove there exist a $p$ so that the inequality holds

I am stuck with the following problem. Given the Gaussian mixture distribution $f(\cdot)$ $$ f(x) = ...
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0answers
48 views

Solution to the Integral

I am trying to solve a pdf which contains the following integral. The integral would involve the inverse of cosine function. Can anybody provide me the method how to solve the below mentioned ...
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3answers
48 views

Integration by parts: $\int e^{-\theta}\cos7\theta \;d\theta$

$$\int e^{-\theta}\cos7\theta \;d\theta$$ I started off by using $u=\cos 7\theta$ and$ \;dv=e^{-\theta}d\theta$, however, this just led me in a circle. I am now at: $$u=e^{-\theta},\;dv=\cos 7\theta ...
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2answers
34 views

Calculus 2: Strategy for Integration, Integral of e^(x+e^x)dx

How would you find $\int e^{x+e^x}dx$? I know I need to use $u$-substitution but I tried changing what I use for $u$ but I still could not get the right answer. If someone could push me in the ...
2
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1answer
43 views

A proof involving nested integrals and induction [duplicate]

Prove that $$\int_0^x dx_1 \int_0^{x_1}dx_2 \cdots \int_0^{x_{n-1}}f(x_n) \, dx_n =\frac{1}{(n-1)!}\int_0^x (x-t)^{n-1}f(t) \, dt$$ I'm trying induction over $n$. The case $n=1$ is trivial. When ...
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1answer
34 views

Integral Test for convergence of a series

"Consider the series given by $$\sum_{n=2}^{+\infty}\frac{1}{n\ln n(\ln(\ln n))^{\alpha}}$$ for $\alpha>1$. Use the Integral Testo to conclude if the series is convergent or not." I tried to make ...
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1answer
27 views

Lebesgue Dominated Convergence: Alternative Proof?

Is there an alternative proof of Lebesgue's dominated convergence theorem relying on positive functions only? The point is I'd like to prove that for positive functions: $$\int ...
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1answer
42 views

How to find $F(x) = \int_x^{x^2} (2+\sqrt t )\, dt$ ?

I have this problem: $$ F(x) = \int_x^{x^2} (2+\sqrt t )\, dt $$ I have to solve the integral. I got $2x^2+\frac{2x^3}{3}-2x-\frac{2x^{3/2}}{3}$ However, I don't think that it correct.
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3answers
236 views

Why we use dummy variables in integral?

I want to know why we use dummy variables in integral? thanks so much.
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2answers
136 views

Why does integration of acceleration data create a slope?

I created a 100hz sine wave in code. When I graph the waveform I get this: When I do an integration on this pure sine wave to get a velocity waveform I get: Is this normal? I do not have a ...
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3answers
63 views

Evaluate the integral $\int_0^{1/4}\frac{x-1}{\sqrt{x}-1}\mathrm dx$

so I have this Integral I have to solve without a calculator. $$\int_0^{1/4}\dfrac{x-1}{\sqrt{x}-1}\mathrm dx.$$ How would I go about finding the antiderivative of that fraction?
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1answer
30 views

Evaluating an integral with unspecified functions $f,g$, given other integrals with these functions

Suppose that $$\int_6^8(3f(x)-x)\,\mathrm dx=6$$ and $$\int_8^6(2x+4g(x))\,\mathrm dx=-8$$ Evaluate $$\int_8^6 (f(x)-5g(x))\,\mathrm dx$$ I have a problem. So, this one question asks me ...
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0answers
15 views

What assumptions should be made?

take a problem like A trough is 12 feet long and 3 feet across. Its ends are isosceles triangles with altitudes of 3 feet. Water is being pumped into the trough at 2 cubic feet per minute. How fast ...
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0answers
45 views

Prove the given two integrals are not equal

I am stuck with following problem: Prove the following two integrals are not equal: $$ \int_{-\infty}^{\infty} p(y-c)\log \big(p(y-c)+p(y+c)\big)dy \neq \int_{-\infty}^{\infty} p(y+c)\log ...
2
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1answer
49 views

If $\int \dfrac{f(x)}{x^2(x+1)^3}\hspace{1mm}dx$ is a rational function, and $f$ is quadratic function, such that $f(0)=1$. Then Find $f'(0)$

If $\int \dfrac{f(x)}{x^2(x+1)^3}\hspace{1mm}dx$ is a rational function, and $f$ is quadratic function, such that $f(0)=1$. Then Find $f'(0)$ This looks like an interesting problem with an elegant ...
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1answer
30 views

Proving an integration with a modified Bessel function and an exponential

I am trying to prove the following identity: where $\mu, h, H$, and $\tilde{\gamma}$ are real constants. The only hint that I have is use the relation between the modified bessel function of the ...
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0answers
55 views

How can I evaluate this integral?? [duplicate]

integral $\int_{0}^{\infty} \frac{cosx}{x^2+1} dx$? I got the answer is $\frac{\pi}{2e}$ by using Wolfram. But can't do it by myself... need some help
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0answers
28 views

Bochner vs. Lebesgue

I'm trying to prove that for complex functions $f:\Omega\to\mathbb{C}$ that are not a priori measurable that: $$f\text{ Bochner integrable}\iff f\text{ Lebesgue integrable}$$ Basically it reduces to ...
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4answers
67 views

What is the most efficient way to integrate $(x-3)\sqrt{x^2+3x-18}$?

I can do the problem, but it is becoming so big,that I do not feel to do it anymore. Can anyone give the shortest method for this problem? $$\int (x-3)\sqrt{x^2+3x-18}\,dx $$
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1answer
34 views

how to remove modulus signs after integrating

$$ \frac{dy}{dt} + k\frac{t^2 -3t + 2}{t+1}y = 0,\ \ \ \ \ \ \ y(t_0=0)=A>0\\ -\int \frac{k}{y} dy = \int (t-4 + \frac{6}{t+1}) dx $$ After integrating the above how do you express $y$ in terms ...