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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12 views

Integral of Normal Distribution with imaginary unit

Hi I need some help with the following integral. $$ \int_{-\infty}^{\infty} \operatorname{e}^{itx} \cdot \frac{1}{\sqrt{2\pi\sigma^2}} \cdot \operatorname{e}^{\frac{-(x - \mu)^2}{2\sigma^2}} dx$$ ...
1
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1answer
21 views

Lower bound on $F$ under the assumption $\theta F(s)\le sF'(s)$

Let $F(s)=\int_0^{s}f(t)\,dt$. We suppose that there exists $\theta>2$ such that $\theta F(s)\le f(s)s$ for all $s\in \mathbb{R}$ and that $F(s)>0$ for all $s\in(0,+\infty)$. How to prove that ...
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0answers
20 views

volume and surface area of elliptic torus

ok, so i have an elliptic torus, parametrised as (i'll just copy the mathematica syntax): rr := {(R + a Cos[v]) Cos[u], (R + a Cos[v]) Sin[u], b Sin[v]} R is here the 'big' radius, and a and b are the ...
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1answer
28 views

Explanation of the passage from $\int_{N'}^N dN/N$ to $\ln N-\ln N'$

While going through my text I got stuck in the derivation given in the picture. ($\Omega$ is a constant) I don't know how to get the second step from the first step, also I don't know why ln is ...
1
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1answer
40 views

Proof of an equation relating definite integrals

The question I have is: Prove that $\displaystyle \int_{0}^{a} f(x) \; dx = \int_{0}^{a} f(a - x) \; dx.$ Since this question occurs at the end of an exercise on integration by substitution, my ...
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0answers
47 views

Equality case in Hölder's inequality

How can I show that $$\left(\int{p(x)^{1-\sigma}dx}\right)^{\frac{1}{1-\sigma}}\cdot \left(\int y(x)^\frac{\sigma-1}{\sigma}dx\right)^{\frac{\sigma}{\sigma-1}}=\int p(x) y(x)\,dx$$ This is to hold ...
1
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2answers
60 views

Indefinite integral of $\frac{2x^3 + 5x^2 +2x +2}{(x^2 +2x + 2)(x^2 + 2x - 2)}$

How do I find $$\int\frac{2x^3 + 5x^2 +2x +2}{(x^2 +2x + 2)(x^2 + 2x - 2)}\mathrm dx$$ I used partial fractions by breaking up $x^2 + 2x - 2$ into $(x+1)^2 - 3$ and split it into $(a+b)(a-b)$ but as ...
0
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1answer
31 views

Integrability condition

Suppose that \begin{align} \mathbb{E}\int_{0}^{T}f^{2}(t)dt <K \end{align} Does it also hold that \begin{align} \int_{0}^{T}f^{2}(t)dt <K \end{align} ? Here, T, K>0 are fixed. I am a bit ...
3
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1answer
48 views

Strange Indefinite Integral

I have a integral which seems difficult to me. Any help would be appreciated. Find $\Large\int \frac{\cos5x + 5\cos3x +10\cos x }{\cos6x+ 6\cos4x + 15\cos2x +10}\large dx$ Also I wound like to know ...
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0answers
17 views

Fitzpatrick's proof of darboux sum comparison lemma

I am just reading Fitzpatrick's advanced calculus. He wants to prove $lim (max(x_{i-1} - x_i)) =0$ and $lim(U(f,P)-L(f,P))$ is equivalent to f is integrable. He used darboux sum comparison lemma, ...
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0answers
22 views

How should I use the integral in this problem?

Let's say there is a charged tube(cylinder with no top or bottom) with radius $a$, length $l$ and charge $q$ and a point which is collinear with the centre of the charged tube. Anyway, since we can ...
0
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0answers
34 views

Quaternion expansion

I have a quaternion equation $ \psi(s)=Pe^{\frac{1}{2}k(s)}\tag 1$ Given conditions and data Here P is a constant unit Quaternion defined for 3D rotation matrix as $(p_1,p_2,p_3,p_4) , p_4\in ...
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votes
2answers
62 views

Integral of inverse of square root of a quadratic

I haven't taken a course on calculus so far so I don't know what to do. The integral may be wrong. Please tell me which part of it is wrong. $$ q∫_{+a}^{-a}\lim_{c \to g}\frac 1{(b^2+c^2)^{3⁄2}} dc $$ ...
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0answers
12 views

how to increace the volume to a specific volume in revolution of solid, using integration [on hold]

two functions, f(x)= 1/9(x-2)^2+7 domain range:{0,10}, g(x)=1/7(x-5)+0.7 domain range: {10,13} increase the volume to 1000ml to 1050mL using integration.
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4answers
55 views

I'm wondering when the integral $\int_{-\infty}^\infty \frac{dx}{(1+x^2)^\alpha}$ converges.

I'm wondering when the integral $$ \int_{-\infty}^\infty \frac{dx}{(1+x^2)^\alpha} $$ converges for the real number $\alpha$.
1
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4answers
46 views

How to prove that $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}dx$ exists

I am trying to show that the integral $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}dx$ exists ($n$ is a natural number). I tried to use the comparison theorem by bounding from above the integrand by ...
0
votes
1answer
53 views

Evaluate the integral with respect to x. [on hold]

I don't know how to begin. Any tips and the correct answer would be appreciated. Evaluate the integral $$\int4^{(x+4^x)}dx$$ Sorry, I read the original problem incorrectly.
5
votes
2answers
83 views

Evaluate $\int\frac{8x+20}{5x^2+25x+20}dx$

I tried to solve it and got $\frac{4}{5} \ln(4+5 x+x^2)+C$ as an answer, but my online homework program says it's incorrect. What did I do wrong? I pulled out $\frac{4}{5}$ as a constant and saw ...
2
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3answers
48 views

Calculate the area of the ellipsoid that rotates around the $x$-axis

So we are about to calculate the area of the ellipsoid around the $x$-axis. $$ \frac {x^{2}}{2}+y^{2} = 1 \implies x=\sqrt{2-y^{2}}$$ We are squaring it so the sign shouldn't matter. I was ...
3
votes
3answers
60 views

Changing order of integration (multiple integral)

Prove $$ \int_0^a\left( \int_0^x \left( \int_0^y \left( \int_0^z f(u) \, du \right) dz \right) dy \right) dx = \int_0^a \frac {(a-t)^3}{3!} f(t) dt $$ where $a$ is constant. So I began with two ...
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1answer
29 views

Solid Angle Integration

Can somebody explain the equivalence between integrating over the surface of a unit sphere and integrating over solid angle? I have been trying to understand the following transformation using a ...
1
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0answers
26 views

Formally evaluating integral to calculate electric or gravitational field.

I never understood how such integrals are calculated, formally. In a line is easy, just a line integral. In a surface, sometimes is easy, like in a disc. But, some surfaces, like sphere, it gets ...
1
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1answer
36 views

Two methods of solving the differential equation $y' = .75 -.005y$

I am working on a differential equation problem and I am stumped since two different methods seem to give me two different answers Method 1 Given $\frac{dy}{dx} = .75 -.005y$ ...
2
votes
6answers
105 views

Proving that $\int \frac{1}{x} dx = \ln(|x|) + c_1$

In all textbooks and online notes, there is always a table of antiderivatives and it always says $\int \frac {1}{x} dx = \ln(|x|)+c_1$ but there is nowhere a proof. I found some proofs online but ...
1
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1answer
55 views

Using integral estimation to show that $ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$

Show with Integral estimation that $$ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$$ $$f(k)=\frac {\ln k}{k^2} $$ For the integral it is : 1 But the other part is the ...
4
votes
2answers
72 views

improper integral containing $\sqrt{\cos x-\dfrac{1}{\sqrt 2}}$ in the denominator

How do i find the value of this integral-- $$I=\displaystyle\int_{0}^{\pi/4} \frac{\sec^2 x \ dx}{\sqrt {\cos x-\dfrac{1}{\sqrt 2}}}$$ I came across this integral too in physics.
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4answers
80 views

Integral of $\int \frac{x\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}dx$

$$I=\int x.\frac{\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}dx$$ Try 1: Put $z= \ln(x+\sqrt{1+x^2})$, $dz=1/\sqrt{1+x^2}dx$ $$I=\int \underbrace{x}_{\mathbb u}\underbrace{z}_{\mathbb v}dz=x\int zdz-\int ...
0
votes
1answer
40 views

Square integrability of functions

Suppose that for a function $f(x)\,\,, x\in\mathbb{R}$ holds \begin{align} \int_{0}^{T}|f(x)|^{2} ~\mathrm{d}x<\infty \end{align} Does it also holds that \begin{align} ...
2
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0answers
49 views

Prove $\int_0^1 \frac{\ln(1+t^{4+\sqrt{15}})}{1+t}dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3})$

Prove that: \begin{equation} \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) \end{equation} ...
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votes
1answer
19 views

Power series function expansion as solution for integral equation

I'm facing an integral equation whose unknown is a function $f(x)$: The equation is of the kind: $$ K = \int_{-l}^{l} G(x,s)f(s)ds $$ So it's a Fredholm integral equation that is rewritten in this ...
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0answers
13 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(t) := \frac{1}{N} ...
2
votes
1answer
127 views

Integral related to a geometry problem

In the question Geometry problem involving infinite number of circles I showed that the answer could be obtained by the sum $$ \sum_{k=0}^{\infty}\int_{B_{k}} {4 \over \,\left\vert\,1 + \left(\,x + ...
1
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3answers
91 views

Formula for $\int {t^n \, e^{t}}dt$?

Consider:$$\int {t^n e^{t}}dt$$ is there any closed formula for this? W|A gave me this but I don't know what is Gamma function: $$\int {t^n e^t dt} = (-t)^{-n} t^n \Gamma(n+1, -t)+ \text{constant}$$
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4answers
59 views

General form for these types of integrals

I encountered this integral in physics-- $$2\int_{0}^{\infty} \dfrac{1-t^2}{(1+t^2)((a+b)t^2+a-b)} dt$$ I know for certain that $a>0$, $b>0$. $a$ and $b$ are independent variables
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0answers
26 views

Complex analysis, cutoff integration

The diff-invariant distance between $z'$ and $z$ is (for short distances) $e^{w(z)}|z'-z|$, so a diff-invaraint cutoff would be at $|z'-z|=\epsilon e^{-w(z)}$. Then $ \int ...
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1answer
40 views

How to integrate $\frac{1}{(x^3-a^3)^3}$ for a few limits

I need to integrate this: $$ \int\frac{1}{(x^3-a^3)^3}\mathrm{d}x $$ For a few limits: $(-a, \infty)$ and for $[0,\infty)$. Just to clarify: $$ \int_{-a + \varepsilon}^{\infty} \text{and} ...
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0answers
70 views

Alternative view on integration?

The question is about an alternative view on formulating or arriving at the concept of the integral (in case this is possible of course). Let's say we want to add a series of values $f(x_i)$ occuring ...
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1answer
35 views

Arc lenght of a curve is finite

Let $b<0<a$, and consider the function $\alpha:(0,+\infty) \to \mathbb R^2$ defined as $$\alpha(t)=(ae^{bt}\cos(t),ae^{bt}\sin(t))$$ Show that $\lim_{t \to +\infty} \alpha'(t)=(0,0)$ and ...
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2answers
50 views

Why is the integral of any orientation form over $\mathbb{S}^1$ non zero?

I am trying to understand the proof of Theorem 17.21 in Lee's Introduction to smooth manifolds; however I am finding myself stuck right at the beginning. The statement I am having trouble with is: ...
1
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1answer
30 views

Volume of a ellipsoidal shape

I was given the following question: My approach so far was to create a parabolic function: y = 25/2 - (25^2)/392 Then I integrate from x = 0 to x = 14 Volume = 2 * pi * integral of y ^ 2 The ...
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0answers
49 views

Evaluating the integral $\int \frac{dx}{\sqrt{a(1+x)^3+1}}$ [duplicate]

How can I solve this integral? $$\int \frac{dx}{\sqrt{a(1+x)^3+1}}$$ where $a>0$. I have tried by using Mathematica, but it fails. Someone has any sugestion?
4
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1answer
66 views

Evaluate $\int\limits_2^4\frac{\sqrt{x^2-4}}{x^2}dx$

Evaluate $$\int\limits_2^4\frac{\sqrt{x^2-4}}{x^2}dx$$ My working: $x=2\sec\theta\quad\Rightarrow\quad\theta=\arccos\left(\frac{2}{x}\right)$ $dx=2\sec\theta\tan\theta d\theta$ ...
1
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1answer
27 views

Solving second order differential equation numerically with values given at intermediate points.

I need to numerically solve the equation, \begin{equation} y''(x) + p(x)y(x) = 1 \end{equation} in the range [a,b] with conditions \begin{eqnarray} y'(\alpha) &=& 1\\ y(\beta) &=& 0 ...
0
votes
0answers
7 views

Histogram Separation Energy Equation

I am working in level set method, specially Lankton method paper. I try to implement Histogram Separation (HS) Energy problem (Part III.C). It based on Bhattacharyya to control the evolution of ...
1
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1answer
19 views

Need help solving an integral for Lagrange Remainder Proof

This image of part of a proof for the Lagrange Remainder for Taylor's Formula. I need help solving the last integral. Can anyone explain?
6
votes
4answers
88 views

Integral of Sinc Function Squared Over The Real Line

I am trying to evaluate $$\int_{-\infty}^{\infty} \frac{\sin(x)^2}{x^2} dx $$ Would a contour work? I have tried using a contour but had no success. Thanks. Edit: About 5 minutes after posting this ...
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0answers
13 views

Simplification of a polynomial before Asymptotic series expansion

I am wondering about a very basic point related to "Asymptotic series expansions". There is a function $f(R)$ which must be expanded as $R$ goes to $ \infty $. Consider that $f(R)=g(R)*p(R)$ where ...
2
votes
2answers
61 views

Evaluate $\int\frac{\sqrt{9-x^2}}{x^2}dx$

I am trying to solve $$\int\frac{\sqrt{9-x^2}}{x^2}dx$$ My answer is slightly different to the memo: Your help is appreciated!
0
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5answers
83 views

Assumptions in Word Problems (Calculus)

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
14
votes
1answer
168 views

Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$ equal to zero?

My initial question was to find if this integral $$ \int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...