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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31 views

double integration with the same variable

I have the integral that I want to resolve. To calculate the flux of the electric machine, I have the following formula: $v_s= R_s \cdot i_s + \frac{\Phi _s}{dt}$ where $v_s, i_s, \Phi _s$ are ...
2
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24 views

Indefinite Hypergeometric Integral Transformations

I'm attempting to solve the indefinite integral $$S\left(v\right) = 2a\sqrt{\alpha ...
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44 views

Gamma Function Representation

I have some questions relating to the gamma function. My question is simply to evaluate the integral $$\int_0^\infty {t^{z-1}\sin t}\,dt$$ whenever $-1<\text{Re}(z)<1$. If we take $z$ ...
2
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41 views

Integration involving exponential function

I am trying to find the following integral $\int_t^s e^{-\frac{(\sqrt{x^2-a^2}-u)^2}{\sigma^2} } e^{-\frac{b^2}{x^2}} dx$ where $t<u<s$, I can find a solution for $u=0$, but I need the ...
2
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62 views

Strict inequality in proof of First Mean Value Theorem for Integrals

Let $M=\sup|f(x)|$ and $m=\inf|f(x)|$ $x \in [a,b]$ The first mean value theoremn for Riemann integrals says: If $f$ continuous (and in this case we will asume non-constant, constant is trivial) ...
2
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51 views

How to integrate $\frac{1}{\sqrt{x^2+y^2+z^2}}$

want to evaluate $$\int\frac{1}{\sqrt{x^2+y^2+z^2}}dxdydz$$ over entire $\mathbb{R}^3$ except $(0,0,0)$. I did this using polar coordinate and got ...
2
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23 views

What is an intuitive/geometric definition of line integrals? Do they work in 2-dimensions?

I understand that we are finding the area of a curve given by some function f(x) over the area of another curve C. (I've also successfully plugged and chugged my way through my homework, without ...
2
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51 views

Differentiation Theorem

Assume that a function $f$ is integrable on $[a,b]$ w.r.t. an increasing function $g$, that $f$ is continuous at $c\in[a,b]$ and that $g$ is differentiable at $c$. Then the function defined by ...
2
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131 views

Confused about trying to find the correct spherical co-ordinates for this tricky triple integral

I'm having trouble trying to figure out how to change the limits of integration to spherical co-ordinates in this particular question. I was wondering if someone would kindly be able to assist me in ...
2
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90 views

Finding $\int \frac{\sin\sqrt{\frac{x}{2}}}{\sqrt{x\cos\sqrt{x}}}dx$

Finding $$\int \frac{\sin\sqrt{\frac{x}{2}}}{\sqrt{x\cos\sqrt{x}}}dx$$ This is a homework. I tried to solve it by assuming $x=u^2$ but after that the integrals become not simple, so I don't know how ...
2
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49 views

On a problem about Rolle's theorem

Let $f:[1,3]\to\mathbb R$ be a continuous function such that $\int_1^2 f(x)dx=2$, and $\int_1^3 f(x)dx=3$, then there exists a real number $c\in(2,3)$ such that $$ \int_1^c f(x)dx=cf(c) $$ Note. I ...
2
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56 views

Double Integral of an Exponential Function with an Absolute Value in the Numerator of the Exponent

This is a question related to statistics, but my major concern relates to the setup and evaluation of integrals. So I decided this question was better suited for Mathematics Exchange than CV. I know ...
2
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37 views

If $A$ and $f$ are bounded, then $f$ is integrable in the extended sense (?) [Spivak]

I have a problem with one of the theorems in Spivak's Calculus on Manifolds. I will give some background first: An open cover $\mathcal{O}$ of an open set $A \subset \mathbb{R}^n$ is admissible if ...
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19 views

An extension of change of variables in double (and $n$-?) integrals - second-order Jacobian?

I'm aware that there are many, many questions regarding changing variables in double and triple integrals. The equation that typically pops up in textbooks is \begin{align} ...
2
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33 views

Proving a result using Riemann-Stieltjes integration

Let $(\alpha_n)_{n=1}^\infty$ be a sequence of monotonically increasing functions on $[a,b]$ such that the series $\sum_{n=1}^\infty \alpha_n(a)$ and $\sum_{n=1}^\infty \alpha_n(b)$ converge. I must ...
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67 views

Is this integral is right or wrong?

We did this exercise in class in a way, but at home I tried to solve it in a different way and I do not know if it is right or wrong. May you help me please? $\mathbf{\int tan^{5}x \, \, \, sec^{4}x ...
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175 views

Improvement of an Inequality

It would nice if someone could help me with this problem. I am looking at an improvement to the classical Jensen's Inequality: $$\int_\limits{}^{} \phi(x) \mu \mathrm{d}x \geq ...
2
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49 views

Prove $\iint_R f_{xy}(x,y)=f(a,c)-f(a,d)+f(b,d)-f(b,c)$

I'm asked to show that if $f_{xy}$ is Riemann Integrable on $R$, then $$\iint_R f_{xy}(x,y)=f(a,c)-f(a,d)+f(b,d)-f(b,c)$$ Where $R=[a,b]\times[c,d]$. To do this I can use the following formulation ...
2
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37 views

Separable equation

I am looking at this first order separable differential equation, and I am stuck. Here is the equation: $\frac{du}{dt}=u$ I seperated like this: $ \frac{du}{u}=dt $ Integrated both sides and ...
2
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37 views

Triple Integral - Varying Order of Integration

I need to express the triple integral for the volume of the solid enclosed by the surfaces $x = 4 - y^2$ and $y + z = 2$ in the first octant in $6$ different ways. I managed to get the correct answer ...
2
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18 views

Evaluate the following integral involving sign function

I = $\int_0^{x_1}\int_0^{x_1}\text{sgn}(y-x) x^{\alpha+i-1}e^{-x/2}y^{\alpha+j-1}e^{-y/2}dx dy$ where, $\text{sgn}(x) = 1$ if $x>0$ and $\text{sgn}(x) = -1$ if $x<0$. Otherwise it is zero.
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30 views

Integration over bounded regions

I have to evaluate the following integral \begin{align*} \int_Ee^{x^2-2xy+10y^2}dA, \end{align*} where $E=\{(x,y)\ |\ x^2-2xy+10y^2\leq1\}$. I already evaluated the integral and got ...
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40 views

Volume of paraboloid

Consider a paraboloid $x^2+y^2+2=z$. The task is to find the volum of the body obtained by confining the paraboloid by several planes $x=0, z=0, y=0$ and $x=3, y=3$. The zero planes cuts out a ...
2
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41 views

compute the complex-valued integral for the branch cut

Let $C$ be the circle of radius $2$ centered at origin. Let $f(z)$ be the branch cut of the function $z^{2−i}$ on the domain $−π < θ < π$. Compute the integral $$ \int_C f(z) dz$$ My attempt: ...
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43 views

Line integral - new parametric equation

We know that $$\int_\gamma V \cdot dr = \int_a^b V(r(t)) \cdot r'(t) dt$$ with $V$being our vector field and $r$being the parametric equation for the curve $\gamma$. Let now $\hat{r} = r \circ \phi$ ...
2
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93 views

$ \int \frac{\sin^{a}x}{\sin^{a}x+\cos^{a}x}dx$ and $ \int \frac{\cos^{a}x}{\sin^{a}x+\cos^{a}x}dx$

I tried solving this integral: $$\int^{\pi/2}_{-\pi/2} \frac{1}{2007^x + 1}\frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx$$ I took a while before aptly applying the following identity I had noted ...
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57 views

Difference between Riemann-Stieltjes and Darboux-Stieltjes integral

Usually, analysis textbooks do not classify Riemann-Stieltjes and Darboux-Stieltjes integral. But I know that a function is $\alpha$-DS integrable does not implies it is $\alpha$-RS integrable. Can ...
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54 views

relationship between $\sin$ and $\cos$ integrals

I am trying to solve a physics problem and I end up with the following two equations. $\int\limits_{0}^{T}\sin\theta(t)dt = a$ and $\int\limits_{0}^{T}\cos\theta(t)dt = b$ The exact dependence of ...
2
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53 views

A tough double integral

I am looking at an ugly ratio of two random variables and am interested in the density of the the ratio. So what I did was to write down the joint characteristic function of the numerator and ...
2
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45 views

Integration imaginary and real part with branch cut

I have some problems with this integral $$ I=\int_{0}^{1}z(1-z)log(1-z(1-z)\frac{q^2}{m^2})dz $$ I see $z(1-z)$ get max value at $\frac{1}{4}$ and if $q^2>4m^2$ log function will be negative and ...
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46 views

closed form for integrals involving error function

Let $F(x)=\int_{-\infty}^xf(t)dt$ where $f(t)=\frac{1}{\sqrt{2\pi}}\exp{(-\frac{t^2}{2})}$. Then is there any way to calculate $$\int_{-\infty}^{\infty}F(x)^kf(x)^2dx$$for $k=2,3,...$ I started with ...
2
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21 views

Solve Intergal Equation of form g.u1=Int(K.u2) for u1 and u2

I'm trying to find a solution to a differential equation of an unusual form: $$g(x) u_1(x)=\int_a^b K(x,y) u_2(y) dy$$ where $g(x)$ and $K(x,y)$ are known and $u_1(x)$ and $u_2(x)$ are complex ...
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78 views

Limit under the integral sign and partition of unity

Let $U \subset \mathbb{R}^N$ be a bounded open set and let $\{ U_j \}_{j=1}^\infty$ be an open covering of $U$ such that $U = \bigcup\limits_{j=1}^\infty U_j$. Suppose $\{ \psi_j \}_{j=1}^\infty$ is ...
2
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60 views

Help with a Proof of exercise 7.3 Apostol on two different definitions of the Riemann integral.

This is a follow-up to this question. Here I ask to check my work and improve the final part that I feel is missing some important steps: So to prove what is asked for in the link above (I am not ...
2
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232 views

Evaluating $\int e^{\Gamma(x)} dx $ and $\int \pi^{\Gamma(x)} dx $

I don't know how to solve these integrals: $$I_1 =\int e^{\Gamma(x)} dx $$ $$I_2 =\int \pi^{\Gamma(x)} dx $$ As a tenth grader I have no idea what the solutions could be. How would one go about ...
2
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80 views

Why does the following Riemann integral exist, but the other doesn't?

By definition if the upper integral equals the lower integral, then $ f $ is Riemann integrable. An example of a Riemann integrable function is $ f(x)=0 $ if $ x\in(0,1] $ and $ f(x)=1 $ if $ x=0 $. ...
2
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0answers
101 views

Residue theorem definite trigonometric integration

I am trying to solve this integration $$\int_{0}^{\pi} e^{cos(\theta)} \tan^{3}(\theta)d\theta$$ putting $$z=e^{i\theta}$$ $$\int_{\gamma} ...
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39 views

Is every Volterra's function unbounded?

Volterra's function is a function $f\colon\mathbb{R}\to\mathbb{R}$ such that: $V$ is differentiable, $V'$ is bounded, $V'$ is not Riemann-integrable. ...
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40 views

Bochner integral vs regulated integral

I'm reading Serge Lang's Real And Functional Analysis and at some point he introduces the regulated integral in order to prove the Fundamental Theorem Of Calculus (in the context of Banach Spaces), or ...
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53 views

Asymptotic expansion of integral of e^(-t)/t^n

So we study $$f_{n}(x) = \int_x^{+\infty} \! \frac{e^{-t}}{t^{n}} \, \mathrm{d}t, \quad n \in \mathbb{N^{*}}$$. I've shown that for every $n$, $f_{n}(x) \sim_{+\infty} \frac{e^{-x}}{x^{n}}$. Now ...
2
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0answers
18 views

How to calculate this integral (with limit afterwards)?

I have to calculate this integral with limit: $$ G(m,n; E) = \lim_{\epsilon \rightarrow 0^+} \iint_{-\pi}^{+\pi} d k_x d k_y \frac{e^{i(k_x m + k_y n)}}{E+ i \epsilon + 2\cos k_x + 2\cos k_y } . $$ ...
2
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122 views

Every Riemann Integrable Function can be approximated by a Continuous Function

Prove that given any Riemann Integrable function $f$ on $[a,b]$, and given any $\varepsilon>0$ one can find a continuous function $g$ on $[a,b]$ such that $$\int_a^b|f(x)-g(x)|dx<\varepsilon ...
2
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0answers
65 views

Numerical integration in 2D over a triangle - Quadrature formula

I am looking for highly (order 6 at least) accurate (for small triangle) quadrature formulas (using only values of the function, no derivatives) to calculate an integral of a continuous function (no ...
2
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22 views

Integration of unknown derivative

I am unable to solve this integral, have forgotten basics and so need help. Shall be very thankful If a way out is provided: $\int_0^R \ln[p'(t)]dN(t) - \int_0^R p'(t) dt$ If $p(t)$ was known then I ...
2
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87 views

A bit challenging integration. (at least for me its challenging)

Hello everybody I am trying to solve this integral. I show you how far I 've gone. $\displaystyle\int^{\infty}_{-\infty} \frac ...
2
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0answers
41 views

If $E(|X|\log|X|)<\infty$ then is $E\left[\frac{|S_n|}{n}\ \log\left(\frac{|S_n|}{n}\right)\right]<\infty$?

I am trying to finish a homework problem in my probability class. I think I am at the end of my problem if I can show that $$E(|X|\log|X|)<\infty$$ implies that $$E\left[\frac{|S_n|}{n}\ ...
2
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0answers
46 views

Area of a region on the surface of a prolate spheroid

Is there a general expression for the area of a region bounded by 3 great ellipses on the surface of a prolate spheroid (where a great ellipse is the intersection of the spheroid with a plane passing ...
2
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0answers
46 views

How to compute the covariance matrix of a random variable uniformly distributed in an ellipsoid

Suppose that x is a random variable uniformly distributed in an ellipsoid \begin{equation} x^{T}Mx\leq\delta, \end{equation} where $x\in \mathbb{R}^{n}$. Clearly, the mean of $x$ is zero. The ...
2
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0answers
67 views

Trapezoidal rule - Multivariable

If I wanted to integrate the function $f(x,y)$ over the region $[a,b]\times[c,d]$ with two segments, am I going about this the right way? $$I(f) = \int_a^b \int_c^d f(x,y)\ dy\,dx = \int_a^b g(x) \ ...
2
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0answers
38 views

Regarding the Lebesgue constant for interpolation

I have a question regarding Lebesgue constant $\Lambda_{n}\left(\boldsymbol{\chi}\right)$, with which the worst case error between an interpolant $p\left(\boldsymbol{x}\right)$ and the function which ...