Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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0
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1answer
36 views

Compute $\int_\gamma\overline{\zeta} \, d\zeta$ using Cauchy’s Integral Formula

Let $\gamma$ be the circle of radius $1$ and centre $0$, equipped with the counterclockwise orientation. Compute $$\int_\gamma\overline{\zeta} \, d\zeta$$ using Cauchy’s integral formula. Any hints ...
0
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2answers
589 views

Volume of a horizontal cylinder using height of liquid

“Tanks” are cylinders with circular cross-section and axis horizontal. These cylinders are variable in size with radius and length different for each tank. We need to determine the amount of liquid ...
0
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1answer
45 views

Derivatives and Integrals of Summations

Im unsure if this is just a stupid question because i have been independently studying this kind of math for about a week, but this has been bothering me lately as i have been exploring some definite ...
8
votes
6answers
223 views

Easiest way to find $\Re\int_{0}^{\pi/2} e^{e^{i\theta}} d\theta$

How do we find $$\Re\left[\int_0^{\large\frac{\pi}{2}} e^{\Large e^{i\theta}}d\theta\right]$$ In the shortest and easiest possible manner? I cannot think of anything good.
1
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2answers
66 views

Fourier Series: ONB

Given the Hilbert space $L^2([-\pi,\pi])$. Consider the orthonormal system: $$\mathcal{S}:=\{\frac{1}{\sqrt{2\pi}}e^{ikx}:k\in\mathbb{Z}\}$$ This is an ONB. How do I prove this? I guess, I could try ...
6
votes
3answers
152 views

Evaluate $\int \frac 1{x^{12}+1} \, dx$

Evaluate $\displaystyle \int \frac 1{x^{12}+1} \, dx$ I tried writing this in partial fractions. $$\int \frac 1{x^{12}+1} \, dx=\int \frac{1}{[(x^6+1)+\sqrt{2}x^3][(x^6+1)-\sqrt{2}x^3]} \, dx$$ ...
-2
votes
1answer
43 views

Show that $\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$

Show that $$\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$$ this is so confusing. i have no idea how to even start. im thinking integration by parts but that ...
8
votes
2answers
95 views

Exactly expressing integral as a sum

Apparently (i.e. according to my professor), the following holds:$$\int_a^b f(x) dx = (b-a)\sum_{n=1}^\infty \sum_{m=1}^{2^n-1} (-1)^{m+1}2^{-n}f(a+m(b-a)2^{-n}).$$How would one go about proving such ...
1
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2answers
71 views

Evaluate $\int_{\partial C} \frac{dz}{(z-a)(z-b)}$ where $\partial C$ is the boundary of a rectangle ($a$ and $b$ are not on $\partial C$)

In discussing the possible outcomes of the integral $$\int_{\partial C} \frac{dz}{(z-a)(z-b)}$$ where $\partial C$ is the boundary of a rectangle ($a$ and $b$ are complex and not on $\partial C$), ...
5
votes
2answers
102 views

Evaluating $\int_{0}^{1}\left(\frac{\ln{(1+x)}}{1+x}\right)^n dx $

I wonder if this integral $$\int_{0}^{1}\left(\frac{\ln{(1+x)}}{1+x}\right)^n dx \quad n=1,2,3,...$$ admits a general formula for integers $n$. I've found $$\int_{0}^{1}\frac{\ln{(1+x)}}{1+x} dx = ...
1
vote
1answer
39 views

$k$-space tensor integral in statistical physics

$$Q=\int_{\text{all space}} \frac{\hbar \nu_g \mathbf{k}\mathbf{k}}{\exp[(\hbar \nu_g |\mathbf{k}|-\mathbf{k}\cdot\mathbf{u})/k_B T]-1}d\mathbf{k} $$ Please help me to integrate the above tensor ...
1
vote
1answer
95 views

Total Mass of a Spherical Object

Consider a spherical galaxy with volumetric mass density, at a distance $s$ from the center, is given by $$ \rho = \frac{k}{1+s^3} $$ where $k$ is a constant. Let $k = 25$. Determine the total mass ...
1
vote
2answers
56 views

How to find the area bounded by $y=\ln\left(x\right)$ and $y=e+1-x$, and the $x$ axis?

Given $\int \int dxdy$, I want to find the area bounded by $y=\ln\left(x\right)$ and $y=e+1-x$, and the $x$ axis. I think the limits of integral in $y$ axis are from $y=\ln\left(x\right)$ to ...
2
votes
0answers
169 views

Interesting problem of finding surface area of part of a sphere.

Show that the surface area of a zone of a sphere that lies between two parallel planes is $2\pi Rh$, Where $R$ is the radius of the sphere and $h$ is the distance between the planes. If you are ...
2
votes
2answers
60 views

$\int _0^1\int _0^{\left(1-x^n\right)^{1/n}}\left(-x^n-y^n+1\right)^{1/n}dydx$

Let $n>0$. How does one integrate $$\int _0^1\int _0^{\left(1-x^n\right)^{1/n}}\left(-x^n-y^n+1\right)^{1/n}dydx$$ ? This integral represents the volume enclosed by ...
0
votes
1answer
34 views

Proving that a solution involving the Laplacian is unique.

I've been asked the following question; If $u$ is a solution of $\nabla^2u = p(x)u$, for $x \in D$, and $\nabla u \cdot n = g(x)$, for $x \in \partial D$, show that $u$ is unique. So, to begin, ...
4
votes
1answer
82 views

Evaluating the integral $ \int_0^{\infty} \cos(x^2)\, \mathrm{d} x$?

Is it necessary to make use of the Gaussian integral and the complex exponential form of the cosine in evaluating the following integral? $$\int_0^{\infty} \cos(x^2)\, \mathrm{d} x$$ Just curious - ...
0
votes
3answers
89 views

Integral using height to find volume

How do you find the volume of a "pit" which is circular in horizontal cross-section, and parabolic in vertical cross-section using height by "sticking". "Sticking" is when we insert a dipstick through ...
3
votes
1answer
77 views

Integration: $\frac{2}{\pi} \int_0^\pi \frac{\sin(nx)}{10}x(\pi-x) \mathrm{d}x$ Can someone matlab/mathematica/maple it for me?

I want to compute $\frac{2}{\pi} \int_0^\pi \frac{\sin(nx)}{10}x(\pi-x) \mathrm{d}x$, where $n=1,2,3,4,\dots$ I worked this by hand and got the result: $-\frac{4\cos(nx)}{10n^3\pi}$ To check my ...
1
vote
1answer
28 views

Does integrating out a variable in a two-variable measurable function produce a measurable function?

This problem is not a mere consequence of Fubini’s Theorem, so I thought that it would be suitable for posting here on MSE. Let $ (X,\Sigma,\mu) $ and $ (Y,\text{T},\nu) $ denote $ \sigma $-finite ...
0
votes
2answers
118 views

Order Statistics Expected Value

Can someone help me with this question? I have arrived with the distribution function equal to $g(y) = \frac{n}{\theta }\left(1-\frac{y}{\theta }\right)^{n-1}$. But couldn't solve the integral for the ...
0
votes
1answer
52 views

(solved)question about proof 3.8 at the book< Measures, Integrals and Martingales> by Rene Schilling?

I am self studying this book having a following question. At page 18 the last line of proof 3.8 says" since every rectangle I is uniquely determined by its main diagonal" then we reach the ...
0
votes
1answer
38 views

Approximate integral via Midpoint Rule

A) Use the Midpoint Rule with $n=6$ to approximate the value of $$\int_0^1 e^{x^2}dx$$ B) Use the error estimate to find the smallest value of $n$ that can be chosen in order to guarantee ...
1
vote
2answers
77 views

Evaluating $\int_0^\infty e^{-x}\cos(x)dx$

By integrating over the contour around an appropriate sector, how does one solve $$\int_0^\infty e^{-x}\cos(x)dx$$
1
vote
1answer
31 views

Complex integral, absolute value of integrand

I want to integrate $f(z)=\frac{1-\mathrm{e}^{\mathrm{i}z}}{z^2}$ over the indented semicircle in the upper half-plane positioned on the $x$-axis as pictured below. The book (Complex Analysis by ...
0
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1answer
45 views

Find the length of the arc of the curve $x= y^5/5 + 1/(12y^3)$ over $[2,4]$

Find the length of the arc of the curve $$x= \frac{y^5}{5} + \frac{1}{12y^3}$$ over the interval $[2,4]$. I know I need to find $$\int_{2}^{4}\sqrt{ 1+\Big(\frac{dx}{dy}\Big)^2} dy$$ because that's ...
1
vote
1answer
46 views

Differential equation of inclined plane

I'm having some trouble with the equation $$\frac{d}{dt}\dot{x}=g\sin\Theta \implies \dot{x}(t)=\dot{x}(t=0)+\int_0^t dt'\:g\sin\Theta=\dot{x_0}+g\:t\sin\Theta $$ which appears in page 4 of ...
2
votes
1answer
99 views

Determine the following integral (very difficult)

So we let $\sqrt{z}$ be the principal value square root of $z$ (i.e. with $\sqrt{1} = 1$ and branch cut along the negative real axis), also let $a \in \mathbb{R}^+$. Determine the following integral: ...
0
votes
1answer
65 views

Show independence of stochastic integral and stochastic process

Let $ M_t $ and $ N_t$ be two continuous local martingales with respect to a filtration $ \mathcal{F}_t $. Suppose that $ M_t $ and $ N_t$ are independent and set $X_t = \int_0^t M_s^4 \mathrm{d} M_s ...
1
vote
4answers
85 views

Why isn't $\int \frac{1}{x}dx$ not defined?

I was thinking about $\int \frac{1}{x}dx$ and how it is defined because if $\int x^{n} dx = \frac{x^n+1}{n+1}$ where $n$ is a constant then: $$\int \frac{1}{x}dx = \frac{x^{-1+1}}{-1+1}$$ $$ = ...
0
votes
3answers
99 views

Example for non-Riemann integrable functions

According to Rudin (Principles of Mathematical Analysis) Riemann integrable functions are defined for bounded functions.For every bounded function defined on a closed interval $[a,b]$ Lower Riemann ...
1
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2answers
111 views

Complex Measures: Integrability

Problem On the one hand, a complex measure decomposes into: $$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$ This gives rise to the integrability condition: ...
0
votes
1answer
76 views

Lebesgue Mean Value Theorem

Disclaimer: This proof is taken out from Rudin, Real and Complex Analysis. Let $\Omega$ be a finite measure space $\lambda(\Omega)<\infty$. Denote the mean value by: ...
3
votes
0answers
28 views

Multiple Integral Substitution Error

I just started learning about the substitution rule for multiple integrals and I decided to give myself an example problem: Calculate $\iint_R{(x^2 + y^2)dA}$ with $R = \{(x, y) \in \Bbb{R} \ |\ 0 ...
9
votes
1answer
332 views

Integration of $\sqrt{x+\sqrt{x^2+3x}}$

I faced following indefinite integration problem: $$\int \sqrt{x+\sqrt{x^2+3x}}dx$$ Result by WolframAlpha suggests that there is an elementary way to compute this integration. But I don't know how ...
11
votes
1answer
410 views

Two integral involving logarithm and polylogarithm function

Evaluate the following integrals $$\int_0^1\frac{\ln(1-x)}{x}\text{Li}_3\left(\frac{1 + x}{2} \right)\,dx\\ .\\ \int_0^1\frac{\ln^2(1-x)}{x}\text{Li}_2\left(\frac{1 + x}{2} \right)\,dx$$
19
votes
4answers
452 views

Prove that $\int_0^1 \frac{{\rm{Li}}_2(x)\ln(1-x)\ln^2(x)}{x} \,dx=-\frac{\zeta(6)}{3}$

I have spent my holiday on Sunday to crack several integral & series problems and I am having trouble to prove the following integral \begin{equation} \int_0^1 ...
1
vote
1answer
47 views

Complex Functions: Integrability

Let $\Omega$ be a measure space with measure $\lambda$. Denote the space of simple functions by: ...
1
vote
1answer
35 views

Why does integration only work. The equation of motion v=u+at gives a different answer.

If given that a particle starts from rest when time t=0. It ravels with acceleration (24t-16)m/s/s. where t is time measured from the instant when the particle is at rest. find its velocity when t=3. ...
1
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1answer
49 views

Radon-Nikodym: Integrability?

Let $\lambda:\Sigma\to\mathbb{R}_+$ and $\kappa:\Sigma\to\mathbb{R}_+$ be finite measures on $\Omega$. Then by Radon-Nikodym: $$\kappa(E)\leq L\cdot\lambda(E)\quad(\forall ...
6
votes
3answers
105 views

How find this integral $I=\int_{0}^{+\infty}\frac{x}{1+e^x}dx$

Question: $$I=\int_{0}^{+\infty}\dfrac{x}{1+e^x}dx$$ I know use ...
0
votes
2answers
23 views

Differentiation of an integral using fundamental theorem of calculus

y = $\int_{0}^{x^2} cos (u^ {\frac {1}{3}}) du $ Find $\frac {dy}{dx}$ My answer y = $3\sin(x^\frac 23)$ $\frac {dy}{dx} = 2cos(x^{\frac23})(x^{\frac{-1}3})$ But by the fundamental theorem of ...
0
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1answer
40 views

How this integration is solved?

Can anyone explain how this integration has been performed? This is a Bayes estimator for uniform prior assuming quadratic loss function. Thanks in advance
7
votes
1answer
135 views

Evaluate $\int_{0}^{1} \frac{\left[\rm{Li}_2\left(\frac{1}{2} \right)-\rm{Li}_2\left(\frac{1 + x}{2}\right)\right]\ln( 1 - x)}{1 + x}\,dx$

$\def\Li{{\rm{Li}}}$How to evaluate the following integral$${\large\int_0^1} {\frac{{\left[ {\Li_2\left( {\frac{1}{2}} \right) - \Li_2\left( \frac{1 + x}{2} \right)} \right]\ln \left( {1 - x} ...
3
votes
1answer
59 views

Complicated integral, where $\int\coth(x)dx$ is somehow written in terms of $\int |x|e^{ix}dx$

In Gardiner's Quantum Noise the following integral equality is used (eq 3.3.10, 3.3.14): $$\int_0^{\infty}d\omega ...
1
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1answer
20 views

Double integral in cylindrical coordinates

I'm having trouble with a double integral problem in cylindrical coordinates. I'm sure the answer is staring me in the face, but I'm missing something. In the multivariable version of the Community ...
0
votes
2answers
46 views

Find the functions $f$ that satisfy the given initial value problems

(a) $f'(x)+3x-2=0$, $f(2)=0$ (b) $2f'(x)-\sqrt{x^3} = 0$, $f(0) = 3$ I know the functions need to be integrated to find $f(x)$, however I am unsure as to how to integrate $f'(x)$ in the ...
1
vote
3answers
100 views

Confirm definite integral equals zero $\frac{\sin(x)}{(1-a\cos(x))^{2}}$

Is this statement about the definite integral of a particular function $F$ true? $$\int_0^{2\pi}F(x)\, \mathrm{d}x = \int_0^{2\pi}\frac{\sin(x)}{(1-a\cos(x))^2}\, \mathrm{d}x = 0 \ \text{ for }\ ...
4
votes
3answers
161 views

Evaluate $\int_{0}^1 x^{p}(\log x)^q dx$

Evaluate $$\int_{0}^1 x^{p}(\log x)^q dx$$ for $p \in \mathbb{N}$ and $q \in \mathbb{N}$.
0
votes
1answer
50 views

Integral of exponent $\iint\limits_0^\infty e^{(t_1 x+t_2 y -y) }\ dy\ dx$

Please help me to solve this equation. I have attempted to answer this one however I always arrived at the wrong answer. Instead of having the positive sign, I always ended with its negative answer. ...