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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4
votes
1answer
34 views

Finding the area under the cycloid $x=t-\sin (t),\;y=1-cos (t)$

I need to find the area under the cycloid $x=t-\sin (t),\;y=1-cos (t)$ above axis and between $x=0,x=2\pi$ using $\underline{\text{Green's theorem}}$ I found in Wikipedia this evaluation: ...
1
vote
0answers
23 views

Can someone help me understand this: integrating over a discrete set of points yields 0 under Lebesgue integral?

Suppose I had some linear function $f(x)$ and then I sampled the function over the integers to form $f(n)$, what would be the evaluation of the Lebesgue integral of $\int_\mathbb{Z_+} f(n) d\mu$? For ...
2
votes
2answers
37 views

Finding the volume of a solid bounded by a sphere and a paraboloid

I am working on a problem that requires me to find the volume of the solid bounded by the sphere $x^2 + y^2 + z^2 = 2$ and the paraboloid $x^2 + y^2 = z$. I know that to do this, I must use triple ...
6
votes
2answers
71 views

Closed-form of $\int_0^\infty \frac{1}{\left(a+\cosh x\right)^{1/n}} \, dx$ for $a=0,1$

While I was working on this question by @Vladimir Reshetnikov, I've conjectured the following closed-forms. $$ I_0(n)=\int_0^\infty \frac{1}{\left(\cosh x\right)^{1/n}} \, dx \stackrel{?}{=} ...
0
votes
1answer
57 views

Evaluate $\oint_{C}xy^2dx+2x^2 dy$

$$\oint_{C}xy^2dx+2x^2y dy$$ triangle:$$C=\{(0,0),(2,2),(2,4)\}$$ My attempt: Using Green's theorem $$\oint_{C}\underbrace{xy^2}_{P}dx+\underbrace{2x^2y}_{Q} dy=\iint\bigg(\frac{\partial ...
5
votes
0answers
22 views

Limit behavior of a definite integral that depends of a parameter.

Let $A>0$ and $1\le \mu \le 2$. Consider a following integral. \begin{equation} {\mathcal I}(A,\mu) := \int\limits_0^\infty e^{-(k A)^\mu} \cdot \frac{\cos(k)-1}{k} dk \end{equation} By ...
1
vote
1answer
20 views

How to find the surface area of revolution of an ellipsoid from ellipse rotating about y-axis

Suppose the ellipse has equation $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$. I understand the way to obtain the surface area of the ellipsoid is to rotate the curve around y-axis and use surface of ...
0
votes
1answer
21 views

Change of variables when integrating over a triangle

I want to calculate the integral $$ \iint_D(x-y)dxdy $$ where D is the triangle made up of the vertices (0,0), (-2,1) and (-1,3). (Graph) My idea was to do this substitution $$ \begin{equation} ...
17
votes
3answers
123 views

Evaluate $\displaystyle\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^\frac{1}{n} \ \mathrm{d}x$

Evaluate $$\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^\frac{1}{n} \ \mathrm{d}x$$ I simplified the limit to $$\dfrac{1}{2}\lim_{n \to \infty} \int_{0}^{\frac{1}{2}} ...
0
votes
0answers
8 views

Error estimate of polynomial quadratures missing some terms

Normally, for trapezoid rule and simpson's rule, etc, error analysis is done by using the error formula for interpolation. However, if the polynomial is restricted to some terms, for example, a ...
-1
votes
0answers
31 views

Quick Integration by parts question [on hold]

If while doing integration by parts I get a sum of +infinity and -infinity, can I obtain that the Integral diverges?
-1
votes
2answers
57 views

Help needed with the integral of an infinite series

Can you please help me with the integral of this series? I came across it in a signal processing paper and haven't been able to figure out the solution myself. $$ ...
5
votes
2answers
90 views

The value of the integral $\int_0^2\left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\:\right)dx$

The value of definite integral $$\int\limits_{0}^{2}\left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\:\right)dx$$ is $$(A)\,4 \quad(B)\,5 \quad (C)\,6 \quad(D)\,7$$ My attempt: I tried using ...
0
votes
1answer
57 views

Evaluate $\iint dydx$ on the domain $0\leq r\leq1$, ${\pi}/{3}\leq\theta \leq{2\pi}/{3}$ [duplicate]

I need to evaluate $\displaystyle\iint \color{red}{dydx}\;\;\;,\bigg\{\frac{\pi}{3}\leq\theta \leq\frac{2\pi}{3}\bigg\}\;\;\;\;,0\leq r\leq1$ $\color{blue}{\text{without using polar ...
0
votes
2answers
56 views

Certain integration technique

What technique to follow when integration functions in the form: $$\sin ax\over \sin bx$$ $$\cos ax\over \cos bx$$ $$\sin ax \over \cos bx$$ I do believe that all these forms should have a similar ...
0
votes
1answer
31 views

Writing line integral as 1-form

If $F: \Bbb R^n \rightarrow \Bbb R^n $ is a vector field and $\phi : [a,b] \rightarrow \Bbb R^n$ is a continously differentiable path we defined the integral of $F$ along $\phi$ as $\int_{\phi} F = ...
2
votes
1answer
37 views

Contour integral of $\frac{1}{\sqrt z}$ with branch cut

I am a physicist who usually doesn't need to care about the fact that square root is not single-valued on the complex plane. But I would like to give a meaning to and compute the contour integrals : ...
0
votes
2answers
39 views

Probability and integration

Compute $E[e^{tX}]$ where $X ∼ \mathcal{N} (0, 1)$. [Hint: Complete the square in the exponent.] Do we set up the integral from $0$ to $1$? Then how do you solve this integral?
11
votes
1answer
280 views

About the integral $\int_{-1}^1 \frac{1}{\pi^2+(2 \operatorname{arctanh}(x))^2} \, dx=\frac{1}{6} $

Here is a question that naturally arose in the study of some specific integrals. I'm curious if for such integrals are known nice real analysis tools for calculating them (including here all possible ...
2
votes
3answers
81 views

$\int_{-\infty}^{\infty}e^{-\pi x^2}\cdot e^{-2\pi ix\xi}dx = e^{\pi\xi^2}$

Prove that for all $\xi \in \mathbb{C}$, $$\int_{-\infty}^{\infty}e^{-\pi x^2}\cdot e^{-2\pi ix\xi}dx = e^{\pi\xi^2}$$ I don't really know how to compute this integral. Can you please help me?
2
votes
1answer
66 views

Finding the area under the curve $y=3-3\cos(t),x=3t-3\sin(t)$

I need to find the area under the curve $\color{blue}{y=3-3\cos(t),x=3t-3\sin(t)}$ and between $\color{blue}{x=2\pi,x=0\text{, above axis}}$ using $\color{blue}{\text{Green's theorem}}$. My attempt ...
6
votes
2answers
121 views

Integrating $\frac{x^3}{\exp(x)-1}$ from $0$ to $\infty$

While doing Physics and trying to prove the law of Stefan-Boltzmann from Plancks-law one comes to the integral \[ \int_0^\infty \frac{x^3}{\exp(x)-1} \mathrm{d}x=\frac{\pi^4}{15} \] and I would like ...
1
vote
2answers
24 views

Finding the flux of $\iint \vec F\hat n\;ds$

I need to find the flux $\displaystyle\iint \vec F\hat n\;ds$ of the vector feild $\vec F=4x \hat i-2y^2\hat j+z^2 \hat k$ throughe the surface $S=\{(x,y,z):x^2+y^2=4,z=0,z=3\}$ My attempt: (I'm ...
2
votes
1answer
37 views

How to prove that $f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt},$ for all invertible functions.

A while ago, I found that: $$f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt}.$$ I managed to prove it for a few functions, and I believe that it may be the case for all ...
1
vote
0answers
33 views

Help solving integration: $I=\int_{-\infty}^{\infty}\phi\left(x\right)\Phi\left(a/\sqrt{b+c\mathrm{e}^{\frac{x-\mu}{\sigma}}}\right)dx$

My work has arrived at needing to solve the integral below for $a,b,c,\sigma>0$ $$I=\int_{-\infty}^{\infty}\phi\left(x\right)\Phi\left(\frac{a}{\sqrt{b+c\mathrm{e}^{(x-\mu)/\sigma}}}\right)dx$$ I ...
1
vote
3answers
84 views

limit of an integral question

Let $f : [0, \infty) \to \Bbb R$ be bounded and continuous. Prove that $\lim \limits _{h \to \infty} h \int \limits _0 ^\infty e ^{-hx} f(x) \, d x = f(0)$. Our intuition was to use l'Hospital's rule ...
1
vote
2answers
48 views

If the area bounded by $y=x^2+2x-3$ and the line $y=kx+1$ is the least, find $k$ and the least area. [on hold]

What concept to use in the Application of Integral question? Please help me
-2
votes
0answers
53 views

Integral calculation question [duplicate]

Calculate the following integral: $\int \limits _0 ^\frac \pi 2 \ln (\sin x) \Bbb d x$. We used the substitution $x=2t$ and then used the identity $\sin 2t = 2 \sin t \cos t$ but now we're stuck. ...
24
votes
2answers
2k views

Why does $ \int_0^1 \lceil { x\sin({1 \over x})} \rceil = 1 - \frac{\log(4)}{2\pi} $?

One time I was bored and played around a bit with integrals and wolfram alpha and tested the following integral: http://www.wolframalpha.com/input/?i=integral_0%5E1+ceil%28x*sin%281%2Fx%29%29 Note: ...
5
votes
2answers
63 views

geometrical interpretation of a line integral issue

I was wondering : if the geometrical interpretation of a line integral is that the line integral gives the area under the function along a path, then why the line integral is equal to zero when the ...
3
votes
3answers
79 views

Evaluate $\iint_{R}(x^2+y^2)dxdy$

$$\iint_{R}(x^2+y^2)dxdy$$ $$0\leq r\leq 2 \,\, ,\frac{\pi}{4}\leq \theta\leq\frac{3\pi}{4}$$ My attempt : Jacobian=r $$=\iint_{R}(x^2+y^2)dxdy$$ $$x:=r\cos \theta \,\,\,,y:=r\cos \theta$$ ...
1
vote
1answer
23 views

Integrating function f(x,y,z) over a rectangular prism

I have a problem that I feel should be fairly straight-forwards, but I do not understand what I'm not doing correctly. $$ f(x, y, z) = x^2 + y^2 + z^2\\ \text{With region: }0\le x\le2\text{ and ...
2
votes
1answer
57 views

How to derive the analytic formula of this integral?

If $F(t) = \int_0^\infty {{{(1 + x)}^t}{e^{ - x}}dx} $, is it possible to get an analytical formula for $F(t)$? Thanks!
1
vote
2answers
26 views

Improper integral convergence question

Prove that the following integral converges: We divided the integral to 2 integrals (one from 0 to 1/2 and the other from 1/2 to 1). We managed to prove that the integral from 1/2 to 1 converges ...
0
votes
1answer
34 views

Is there any way to combine the product of two univariate integrals into a single integral?

Can two separate integrals but multiplied together in the end by integrated as a product once instead? In other words, does $$ \left(\int_{-\infty}^{+\infty}f(x)\mathrm{d}x ...
-1
votes
2answers
77 views

Improper rational/trig integral comes out to $\pi/e$ [on hold]

During my studying to integration I find this integration. So I tried to prove but I got stuk. So I need help in this integration. $$\displaystyle\int_{-\infty}^{\infty} \frac{x \sin (x)}{1+x^2} ...
2
votes
3answers
89 views

Compute the integral

Compute the integral: $$\frac{1}{2\pi i}\int_{|z| = 1}\frac{(z-b)^m}{(z-a)^n}dz$$ where $|a| < 1 < |b|$; $m, n \in \mathbb{Z}$ My approach is using Cauchy integral formula, we have ...
-4
votes
2answers
44 views

How we can prove this??? [on hold]

I study integration and find this... need to know this integration How come? $\displaystyle\int_{0}^{\infty} \frac{e^{-x} - e^{ -2x}}{x} dx = ln (2) $
3
votes
2answers
43 views

compute the integral $\int_{|z|=1}\left[\frac{z-2}{2z-1}\right]^3dz$

Compute this integral: $$\int_{|z|=1}\left[\frac{z-2}{2z-1}\right]^3dz$$ my solution is I used derivative of Cauchy integral formula, which is $$f^{(n)}(z_0) = \frac{n!}{2\pi i}\int ...
3
votes
3answers
158 views

Definite integral with limits from zero to infinity

Let $ I=\int\limits_{0}^{\infty}e^{-(x^2+\frac{1}{x^2})}dx$ and $J=\int\limits_{0}^{\infty}x^2e^{-(x^2+\frac{1}{x^2})}dx$. If $J=\dfrac{pI}{q}$, then find the value of $p+q$ where $p$ and $q$ are ...
5
votes
2answers
57 views

A problem related to integration in $L^1$

If $f\in L^1[0, 1]$ and $\int_{0}^1 x^nf(x)=0$ for all $n = 0,1,2,...$then prove that $f$ is identically zero almost everywhere. This would be very easier to prove if $f$ were continuous on $[0, 1]$ ...
5
votes
3answers
210 views

Not the toughest integral, not the easiest one

Perhaps it's not amongst the toughest integrals, but it's interesting to try to find an elegant approach for the integral $$I_1=\int_0^1 \frac{\log (x)}{\sqrt{x (x+1)}} \, dx$$ $$=4 ...
1
vote
4answers
88 views

Evaluate $\iint dy\,dx;\frac{\pi}{4}\leq\theta \leq\frac{3\pi}{4};0\leq r\leq2$

I need to evaluate $\displaystyle\iint \color{red}{dydx}\;\;\;,\frac{\pi}{4}\leq\theta \leq\frac{3\pi}{4}\;\;\;\;,0\leq r\leq2$ $\color{blue}{\text{without using polar coordinates}}$. My attempt: ...
2
votes
0answers
97 views

Can $\int_{0}^{1}\frac{x^{p}\ln^{q}(x+a)}{(x+a)^{b}}dx$ be expressed in a simple form?

I was browsing the book Irresistible Integrals and found this gem, at page 97, $$ \int_{0}^{1}x^{n}\ln^{k}(x)dx=\frac{(-1)^{k}k!}{(n+1)^{k+1}} $$ that resembles a previous question of mine here. So, ...
1
vote
0answers
37 views

Integral with Bessel Functions

Any suggestions how to solve this: \begin{equation} \int_0^a\int_0^\infty J_0 (\lambda r)J_1(\lambda a)\frac{1}{\sqrt{n+\lambda^2 }}d\lambda dr \end{equation} (J0,J1 Bessel function of first kind, ...
2
votes
4answers
43 views

Use comparison test to determine convergence

$$\int_{1}^{\infty}\frac{\ln x}{\sinh x}dx$$ I tried several functions and failed to get integrable convergent bigger function. Thanks for help.
0
votes
1answer
32 views

Evaluate $\int_{-2}^{2}\int_{y^2-3}^{5-y^2}dxdy$ [duplicate]

In the black I evaluated the integral and I got 64/3, now I need to evaluate the same integral with $\color{red}{dydx}$ .in the $\color{blue}{\text{blue}}$ color is my attempt, I don't think that my ...
-4
votes
0answers
25 views

integrate very long expression using orthogonality in maple [on hold]

I have very long expression and i must integrate it. i try to apply "orthogonality" on my equations to eliminate "X" and "Y" variables. Image Shows examples of Orthogonal Functions my integral code ...
2
votes
2answers
53 views

A question on use of square integrable functions

I'm approaching this from a physicist's perspective, so apologies for any inaccuracies (and lack of rigour). As far as I understand it, a square-integrable function $f(x)$ satisfies the condition ...
2
votes
1answer
17 views

Properties of unimodal functions

A probability density function $f$ is said to be unimodal if the value at which the maximum value of the function is attained is unique. I am reading some papers on econometrics that appear to use ...