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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5
votes
2answers
81 views

If $\int_0^\infty x f(x) \ dx < \infty$, is $\int_0^\infty \int_0^\infty f(x+y) \ dx \ dy< \infty$?

Question: Let $f : [0, \infty) \to [0,\infty)$ be a measurable function which, more than being integrable, satisfies $\int_0^\infty x f(x) < \infty$. Does it follow that $\int_0^\infty ...
0
votes
1answer
34 views

Evaluate the integral

Everyone, I am try to evaluate this integral for one day, could anyone give me some hints and help me solve this integral? $$ \int_0^{\infty}\frac{r}{1+s^{-1}[r^{-\alpha}+{(r+d)}^{-\alpha}]^{-1}}dr $$ ...
7
votes
3answers
102 views

Calculation of integral $\int\exp \left(-\alpha \sin^2 \left(\frac{x}{2} \right) \right) dx$

Given $\alpha$ is a constant. How to calculate the following integral? \begin{equation} \int \exp \bigg(-\alpha \sin^2 \bigg(\frac{x}{2} \bigg) \bigg) dx \end{equation} Thanks for your answer.
1
vote
0answers
33 views

Can one obtaining a mean value form of the Taylor series remainder using the integral remainder?

Can we show that $$(\exists \epsilon \in[0,x])\left(\int_{0}^x \frac{(x-s)^n f^{(n+1)}(s)}{n!}ds= \frac{x^{n+1}f^{(n+1)}( \epsilon)}{k!}\right)\text{ ?}$$ Thanks in advance!
5
votes
1answer
56 views

integral involving hypergeometric function $\int^1_0\frac{_2F_1(p,p;p+1;-\frac{1}{y})}{y}\,dy$

I arrived at the following result $$\tag{1}\int^\infty_0 z^{p-1} E^2(z)\,dz=\frac{\Gamma(p)}{p}\int^1_0\frac{_2F_1(p,p;p+1;-\frac{1}{z})}{z}\,dz$$ where the exponential integral $E(z)$ is defined ...
0
votes
2answers
68 views

Definite integral versus indefinite integral evaluation

Why evaluating $$\iint x\, \mathrm dx\, \mathrm dx$$ in $[0;2]$ is different from calculating $$\int^2_0 \int^2_0 x\, \mathrm dx\, \mathrm dx$$ ? What is the conceptual difference between the two?
2
votes
1answer
70 views

Line integral: $u = ( \frac{-y}{x^2 + y^2}, \frac{x}{x^2 + y^2}, z)$

Let $u = ( \frac{-y}{x^2 + y^2}, \frac{x}{x^2 + y^2}, z)$ and D the domain bounded by the torus obtained by rotating the circunference $(x-2)^2 + z^2 =1, y=0$ around the z-axis. Show that $rot( u )=0 ...
0
votes
0answers
20 views

Maximum Principle - Proof

We want to show the maximum principle for a function $f = f(x,t)$ on a n-dimensional hypersurface $M,$ that is, (Corollary) Let $f = f(X,t)$ be a function on M, let $\vec{a}$ be a vector field on ...
0
votes
1answer
35 views

fourier series representation

Find the Fourier series with period $2$ of $$f(x) = -x,\qquad-1<x<1$$ so I find that $a_0$ and $a_n$ both are $0$ since odd functions so the Fourier series is on the form: ...
1
vote
1answer
69 views

Evaluate the integral of primitive $\frac{1}{x(x+2)}$

I am doing this integral by example: $$\int_1^\infty\frac{1}{x(x+2)}\ dx.$$ The example in the book, starts with $\dfrac{1}{x(x+2)} \leq \dfrac{1}{x^2}.$ Why is this important? Why does the example ...
0
votes
1answer
197 views

Finding an area of a propeller using double integration? Attempted, please help! :( [duplicate]

The boundary of the internal hole is given by $r = a + b\cos(4α)$ where $a > b >0$. The external boundary of the propeller is given by $r = c + d\cos(3α)$ where $c - d > a + b$ and $d > ...
8
votes
1answer
78 views

Need help with a definite integral [duplicate]

Evaluate: $$\int_0^{\infty} \frac{x-1}{\sqrt{2^x-1}\ln(2^x-1)}\,dx$$ I am not sure where to start or what should be the best approach towards this problem. I tried the substitution $2^x-1=t^2$ but ...
0
votes
1answer
26 views

inequalities for monotonic functions

Let $f,\tilde{f},g,\tilde{g}\colon\mathbb{R}\to[a,b]\subset\mathbb{R}_{\geq 0}$ be positive, continuous and increasing functions, with $f\geq \tilde{f}$ and $g\geq\tilde{g}$. Does it hold that ...
4
votes
1answer
49 views

Finding the area under a curve represented by the equations $x=a\cos{t}+\frac{a}{2}\ln{\left(\tan^2{\frac{t}{2}}\right)}$ and $y=a\sin t$

How do I find the area of the curve represented by the following equations, $$x=a\cos{t}+\frac{a}{2}\ln{\left(\tan^2{\frac{t}{2}}\right)}\\ y=a\sin t$$ Here's what I tried: Let $A$ denote the area ...
5
votes
1answer
75 views

Is there a useful relationship between pointwise and $L^2$ distance?

It would be really convenient to get a bound on the point-wise closeness of functions by knowing their $L^2$ distance. Clearly, if two functions are close in the $L^2$ sense, you cannot get a general ...
1
vote
4answers
117 views

Double integral for $\int_{0}^{1} \int_{-1}^{0} \frac {xy}{x^2 + y^2 + 1}\ dy\ dx$

I'm trying to evaluate this $$\int_{0}^{1} \int_{-1}^{0} \frac {xy}{x^2 + y^2 + 1}\ dy\ dx$$ tried substition $$ u = {(x^2+y^2+1)}^{-1} \ \ du = \ln {(x^2+y^2+1)}$$ but du is not found in the ...
0
votes
3answers
147 views

Integration of $x/\sqrt{x^2-7}$ using trigonometric substitution

Didn't find this one here so I'm asking away: I tried integrating by substitution by ended up with just $x + C$ which is clearly wrong. My work, as requested:
0
votes
0answers
31 views

polar co ordinates integration

integrate the polar co ordinates $$ \int^{r=\infty}_{r=0} \int^{z=\infty}_{z=-\infty} \delta(r) \delta(z-z_s) dz dr$$ => I want to integrate the above equation. integral of $ \int^ {z=\infty} _ {z= ...
0
votes
2answers
74 views

How to find the limits of a triple integral converted to spherical coordinates

Find the integration limits of $\int_{0}^{3} \int_{0}^{\sqrt{9 - x^2}} \int_{0}^{\sqrt{9 - x^2 - y^2}} \frac{\sqrt{x^2 + y^2 + z^2}}{1 + (x^2 + y^2 + z^2)^2} dz dy dx$ in spherical coordinates. ...
0
votes
1answer
40 views

How do I find the limits for $\iiint_{W} \frac{dx dy dz}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}$?

Evaluate $\iiint_{W} \frac{dx dy dz}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}$ where $W$ is the solid bounded by the two spheres $x^2 + y^2 + z^2 = a^2$ and $x^2 + y^2 + z^2 = b^2$ where $0 < b < a$. ...
1
vote
0answers
78 views

Improper integral calculation and nature

I would like to know how to calculate this integral n check its nature : $$ A= \int_1^\infty \frac{e^{-t}}{1+t^{2}} dt . $$ I think I figured it out , we can do it with Equivalence like this (since ...
1
vote
0answers
52 views

How to solve Max under an integral?

This is the first time I come accross a Max function inside an integral. I have looked around online and did not find anything about it. I would like to know the rules of what can I do when I have an ...
2
votes
0answers
78 views

Geometrically Integrating $R(x,\sqrt{Ax^2+Bx+C})$ & Motivating Euler Substitutions

I believe there is a way to geometrically interpret integrating $R(x,\sqrt{Ax^2+Bx+C})$, as a means to motivate the Euler substitutions, in terms of "...expressing the coordinates of a point upon ...
2
votes
0answers
46 views

Finding flux of a vector field $\mathbf{F}$ across a surface bounded by an unknown function.

A solid $\Gamma$ in $\mathbb{R}^{3}$ is bounded by $$0 \leq x \leq 1, \hspace{.5cm} 0 \leq y \leq 1, \hspace{.5cm} 0 \leq z \leq g(x, y),$$ where $z = g(x, y)$ is an unknown differentiable surface. ...
2
votes
1answer
145 views

Show that $f\colon\bar{M}\to\mathbb{R}$ is Riemann-integrable

Consider a Jordan-measurable set $M\subset\mathbb{R}^n$ and $f\colon\bar{M}\to\mathbb{R}$ continious. Show that $f$ is Riemann-integrable over $M$. I think the main facts for the prove ...
1
vote
1answer
68 views

calculating the absolute value integral

I would like to know how to calculate this integral $$ B= \int_0^1 \mid 1- t^{a} \mid^{b} dt . $$ I know for $$ a>0 $$ $$ B= \int_0^1 (1- t^{a})^{b} dt . $$ and for $$ a<0 $$ $$ B= ...
4
votes
1answer
118 views

Problematic integral $\int_0^\pi \frac{x\sin x}{1+\cos^2x}\ dx$

How to calculate $$\int_0^\pi \frac{x\sin x}{1+\cos^2x}\ dx\ ?$$ I wish I could say I ran out of ideas, but actually I have none.
2
votes
1answer
38 views

Switch integral and differential operator

I want to know if $$ \frac{d}{dt} \int_{-\infty}^\infty e^{\large- \frac{x^2}{c+2t}} \, dx = \int_{-\infty}^\infty \frac{d}{dt} e^{\large- \frac{x^2}{c+2t}} \, dx $$ holds. Therefore I have to ...
2
votes
0answers
23 views

A proof regarding Fourier-Polynoms

I want to prove the following: Let $f:\mathbb{R}\rightarrow \mathbb{C}$ so that $f \big |_{[0,2\pi]}$ is integrable. Let $V$ be the vectorspace of all $2\pi$-periodic functions and $U \subset V$ be ...
0
votes
0answers
21 views

Jordan content under continuous differentiable map

I have the following problem which seems simple but in fact I find no proof for it so I am wondering if I could get some help. Let $A$ be a compact set subset of an open set $U$ in $\mathbb{R^n}$, ...
7
votes
3answers
110 views

Integration of $\int_{0}^{\frac{1}{2}}\frac{\sin^{-1}(x)}{\sqrt{1-x^2}} dx$ ??

I was solving the integration of inverse trigonometric function and faced a question which i find it hard to understand. I need to find the definite integration of this function. ...
6
votes
1answer
132 views

A solution for $\int^{2\pi}_0e^{\cos \theta}\cos(a\theta -\sin \theta)\,d \theta $

It can be proved using complex analysis that $$\tag{1}\int^{2\pi}_0e^{\cos \theta}\cos(n\theta -\sin \theta)=\frac{2\pi}{n!}$$ My initial thought that, we use the Gamma function for non-integer ...
4
votes
3answers
106 views

Calculate $\int \frac{1}{x^2+x+1}\mathrm{d}x$

Define the integral $I$ as follow: $$I=\int \dfrac{1}{x^2+x+1}\mathrm{d}x.$$ I do not know how to integrate it. Any suggestions please? I tried a lot of methods: I substituted $x^2+x=u$. I ...
0
votes
0answers
51 views

moment generating function of a shot noise process

A general setting shot noise process $X(\tau)=\sum \limits_k Z(\tau, T_k)$ where $T_k$ Poisson process with intensity $\lambda(t)$, $Z(.,t)$ independent stochastic processes. Show that the moment ...
2
votes
1answer
57 views

Cylindrical cordinates: $\iiint (x^2 + y^2 + z^2) dxdydz$

Show that $$ I= \iiint_S (x^2 + y^2 + z^2) dxdydz = \frac{2^{10} a^5 k}{75} \left(1 + \frac{k^2}{3} \right), a>0, k>0$$ where $S$ is the region bounded by the cilinder $x^2 + y^2 = 2ax$ and ...
15
votes
4answers
339 views

For which $n$ is $ \int_0^{\pi/2} \frac{\mathrm{d}x}{2+\sin nx}= \int_0^{\pi/2} \frac{\mathrm{d}x}{2+\sin x}=\frac{\pi}{3\sqrt{3\,}\,}$?

I have been trying to figure out for which $n$ is $$ \int_0^{\pi/2} \frac{\mathrm{d}x}{2+\sin nx} = \int_0^{\pi/2} \frac{\mathrm{d}x}{2+\sin x}=\frac{\pi}{3\sqrt{3\,}\,}$$ Using maple I got the ...
0
votes
1answer
37 views

Formula for area under the curve

I don't know that the equation that I am going to explain below is correct or not, and this is why I am asking this question. So, I have found out that area under the curve could be found out by ...
3
votes
2answers
64 views

Integral of $\sqrt{a^2+b^2t^2}$

I'm trying to calculate mass of some line and this is the integral needed to be solved. Wolfram shows me some way with the fuction sec and reduction methods and I don't know how to use these. is ...
1
vote
2answers
71 views

Integral of exponential product function

I want to know the value of this integral: $$\int_{0}^{\infty}e^{-u}e^{-u^{\alpha}x}\mathrm{d}u$$ where $\alpha>0$, $x>0$. Thank you.
4
votes
3answers
84 views

How to integrate $\int_0^{\pi/2} \ \dfrac{\cos{x}}{\sqrt{1+\cos{x}}} \, \mathrm{d}x.$

I need to somehow evaluate the following: $$ \int_0^{\pi/2} \ \dfrac{\cos{x}}{\sqrt{1+\cos{x}}} \, \mathrm{d}x. $$ Can anyone give me any hints/pointers? I've tried to use parts, and some feeble ...
3
votes
2answers
69 views

Calculating the Integral

I would like to know how to calculate this integral $$ A= \int_0^1 \ln(1-t^{a}) dt . $$ I tried Taylor expansion for $\ln(1-t^{a})= -t^{a}$ , that gave me this : $$ A= \lim_{ x \rightarrow 0+} ...
2
votes
1answer
34 views

Integration by subtitution

Can someone explain me how to find the value of $$L = \int_1^2 \sqrt{1+9x} \,\mathrm{d}x$$ I do not know how to approach it after having $z = 1+9x$ and $\mathrm{d}x = \mathrm{d}z/9$.
0
votes
1answer
19 views

double integral domain setting

Doing a double integral question always involve transfer the domains into $a<x<b$ and $ f(x)<y<g(x)$ but sometimes, it's very hard to find these ranges Like $|x|+|y|\leq 1$ or $x\leq ...
2
votes
1answer
48 views

Does the change of variable function have to be injective?

Please note that I'm only interested in the one-variable case here. The change of variables formula for integration is: $$\int^{\phi(b)}_{\phi(a)}f(x)\ \text{d}x= \int^b_a f(\phi(x))\phi'(x)\ ...
2
votes
2answers
214 views

Area of Intersection of Circle and Square

Given a point $(x,y)\in [0,1]^2$ and $r > 0$, I would like to derive a general formula for the area of the intersection of the circle of radius $r$ centered at $(x,y)$ and the unit square. What is ...
1
vote
0answers
57 views

What will be the integration region?

Where $\Omega_s$ is new integration region, due to change in geometry integration region will also change. Also note that $\Omega_l$ is $\Omega$ with $|x|<l$ is the integration region for ...
0
votes
1answer
53 views

volume of solid by rotating the region by given curves

Which of the following integrals represents the volume of the solid obtained by rotating the region bounded by the curves x^2 - y^2 = 7 and x= 4 about the line y = 9? A. ∫ from -3 to 3 2π (y - 9) ...
0
votes
1answer
35 views

Finding $\iint_D \nabla \cdot F \; \mathrm{d}A$

I want Finding $\iint_D \nabla \cdot F \; \mathrm{d}A$. Normally I have dealt with $\nabla F$ for $F(x,y,z)=xyz$ sort of cases, where I just then derive it in terms of $x,y,z$ for my $i,j,k$. Here ...
3
votes
1answer
62 views

Compute $ \int\sin(x^2)\, dx + \int \sqrt{\arcsin t}\, dt$

Compute the sum of two integrals $$ \int_{\large\sqrt{\frac{\pi}{6}}}^{\large\sqrt{\frac{\pi}{3}}}\sin(x^2)\ \ dx + \int_{\large\frac{1}{2}}^{\large\frac{\sqrt{3}}{2}} \sqrt{\arcsin t}\ \ dt. $$
3
votes
2answers
116 views

Evaluate $\displaystyle\int{\frac{e^{2x}} {\sqrt{1-e^x}}}\ dx$

Evaluate $$\displaystyle\int{\frac{e^{2x}} {\sqrt{1-e^x}}}\ dx.$$ I tried to solve by using integration by parts, but I couldn't find a solution. What method should I use to integrate this?