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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13 views

An application $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ and $C^{1}$ such that $f(x)=0$ for $x>r$ implies the value of jacobian integral is zero

Let $f \colon \mathbb{R}^n \to \mathbb{R}^n$ of class $C^{1}$. Suppose that exists $r>0$ such that $f(x)=0$ if $|x|\geq r$ .Prove that exists $k>0$ such that: $\displaystyle \int_{B[0,k]}$ det$...
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1answer
13 views

What to write when computing double integrals in order to minimize errors

This is a bit of an odd question, but I am learning how to compute lots of calc things from books so I don't have the benefit of watching how a teacher does it. I tend to be very error prone when ...
4
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2answers
195 views

How to integrate $\int \dfrac{x^{13}\ dx}{x^5 + 1}$

We get this problem from our teacher today. I only wish that it was $x^{14}$ in the numerator, so we can use substitution method: $$\int \dfrac{x^{13}\ dx}{x^5 + 1}$$ I cant find way to integrate ...
0
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1answer
31 views

Difficulty setting up an iterated integral

I am trying to integrate the function $\frac{1}{\sqrt{2y-y^2}}$ over the region in the first quadrant bounded by $x^2=4-2y$. Given that this region is between bounded by an convex parabola and in the ...
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1answer
23 views

Find the depth of the propane in the tank when it is filled to one-quarter of the tank's volume.

I'm new to this site. My boyfriend has some calculus problems that he's unable to complete due to a family emergency, and so I am trying to help him. His professor hasn't emailed him back about ...
4
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1answer
55 views

Evaluation of $\int_{0}^{\sqrt{2}-1}\frac{\ln(1+x^2)}{1+x}dx$

Evaluation of $$\int_{0}^{\sqrt{2}-1}\frac{\ln(1+x^2)}{1+x}dx$$ $\bf{My\; Try:::}$ Let $$I(a) = \int_{0}^{\sqrt{2}-1}\frac{\ln(1+ax^2)}{1+x}dx$$ Now $$I'(a) = \int_{0}^{\sqrt{2}-1}\frac{x^2}{(1+ax^...
2
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0answers
38 views

Dumbbell Contour? $\int_0^1 \log(x)\log(1-x)dx$ via complex methods.

Having evaluated this integral via the power series and various approaches via special functions, I'm now curious if there is a direct way to compute this integral by taking a slit along $[0,1]$ and ...
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1answer
16 views

Find the volume of the solid generated by rotating about the x-axis the region bounded by the curves

I'm new to this site, and this will be my last question post. My boyfriend has a few problems that he's unable to complete because of a family emergency and I have decided to try and help him. His ...
3
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1answer
21 views

Finding the volume of a solid generated by revolving around the line y=6

I'm new to this site. My boyfriend is unable to complete these problems due to a family emergency and I was going to try to help him because they are due in a few hours and his professor hasn't ...
0
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1answer
67 views

Is it true that $\int{\frac{\partial}{\partial t}f(x,t)\,\mathrm dx} = \frac{d}{dt}\int{f(x,t)\,\mathrm dx}$?

I was wondering why and when is true that: $\displaystyle\int{\dfrac{\partial}{\partial t}f(x,t)\,\mathrm dx} = \dfrac{d}{dt}\int{f(x,t)\,\mathrm dx}.$ I would appreciate any help.
3
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1answer
58 views

How do I prove this $\int_{0}^{\infty}{e^{-x^n}-e^{-x^m}\over x\ln{x}}dx={\ln{\left(m\over n\right)}}?$

How do I prove this $$\int_{0}^{\infty}{e^{-x^n}-e^{-x^m}\over x\ln{x}}dx=\color{blue}{\ln{\left(m\over n\right)}}.\tag1$$ I know of the standard integral $$\int_{0}^{1}{x^m-x^n\over \ln{x}}dx=\...
19
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4answers
968 views

Looking for closed-forms of $\int_0^{\pi/4}\ln^2(\sin x)\,dx$ and $\int_0^{\pi/4}\ln^2(\cos x)\,dx$

A few days ago, I posted the following problems Prove that \begin{equation} \int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}\\[20pt] -\int_0^{\pi/2}\ln^3(\cos x)\,dx=\frac{\pi}{2}...
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0answers
17 views

Integral of magnetic field inside cylinder

Let $V\subset\mathbb{R}^3$ be an infinitely high solid cylinder, or a cylindrical shell of radii $R_1<R_2$, whose axis has the direction of the unit vector $\mathbf{k}$. For any point of ...
9
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2answers
619 views

A slightly problematic integral $\int{1/(x^4+1)^{1/4}} \, \mathrm{d}x$

Question. To find the integral of- $$\int{1/(x^4+1)^{1/4}} \, \mathrm{d}x$$ I have tried substituting $x^4+1$ as $t$, and as $t^4$, but it gives me an even more complex integral. Any help?
2
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0answers
28 views

A sufficient condition on a real smooth function

Let $f : [0, \infty) \to \mathbb{R}$ be a smooth function. I would like to find a sufficient condition on $f$ in order to have that $$ \liminf_{t\rightarrow \infty} \int_0^t \Big(\frac{t - s}{s} \Big)...
0
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1answer
23 views

Rotaion Surfaces and Complex Numbers

Consider a continuous invertible map $\varphi:\mathbb{R}^+ \longrightarrow \mathbb{R}$, and define the follwing surface $$ s:\mathbb{C} \longrightarrow \mathbb{R} \times \mathbb{C} $$ $$ \qquad xe^{i\...
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3answers
70 views

Why is dividing by $dx$ or some other differentitator in an integral considered taboo?

Forgive my ignorance but I remember when I was learning calculus, I remember that when we integrate, we always multiply the differentiator to $F(x)$. However, it was never explained to me why we ...
0
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1answer
52 views

integration of exp(cos(x-a)) dx

I would like to compute either of the following integrals: $$\int e^{\cos(x-a)} \, dx$$ or $$\int_{-\pi}^{\pi} e^{\cos(x-a)} \, dx$$ In both cases, $a$ is a constant. MATLAB doesn't seem to be ...
1
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1answer
37 views

Evaluate the integral $\int_c^{\infty}(1-e^{-abc^{m}x^{-m}d})xdx$

I need to know how to solve the following integration $$\int_c^{\infty}(1-e^{-abc^{m}x^{-m}d})xdx$$ where $a,b,c,d,m$ are all greater than $0$. I will be very grateful for your help.
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0answers
38 views

Is it possible to find an approximate or an exact answer for this integral?

I am trying to solve this integral $$ \int\limits_{x_{0}}^{\infty}\frac{2}{x \sqrt{\frac{x^2 }{x_{0}^2}\Big(1-\frac{1}{x_{0}}\Big)-\Big(1-\frac{1}{x}\Big)+\epsilon \left((x-1)^{\nu }-\frac{x^2 }{...
1
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0answers
44 views

An integral with the Gamma function

Let $c$ be a positive constant. Can you evaluate the following integral? $$ \int_0^{\infty} \frac{x^{cx+1}}{\Gamma(1+cx)}\mathrm{e}^{-x}\,\mathrm{d}x $$ Ps. Do you have some advices or tricks to ...
0
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0answers
51 views

area of a rectangle

I know that to use polar coordinates instead of $dxdy$ we have $dA=rdrd\theta$. As such, we can have a double integral like $$ \int_{\theta=a}^{\theta=b}\int_{r=c(\theta)}^{r=d(\theta)}f(r\cos\theta,r\...
0
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0answers
18 views

Any simplifications possible for the following integral?

Here is an integral $$\int_0^a\mathop{\mathrm{d}z}f(z)\mathrm{e}^{\int_0^zf(z')\mathrm{d}z'+\int_0^zg(z')\mathrm{d}z'},$$ where all functions are real and positive and $a>0$. Is there any ...
0
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2answers
73 views

Evaluate $\iint { \sqrt{\left| y-{ x }^{ 2 } \right|}\, dx\,dy } $ over a rectangle

Question: I want to evaluate $\iint_R {\sqrt{ \left| y-{ x }^{ 2 } \right|}\, dx\,dy }, $ where $R=[-1,1]\times[0,2]$. Indeed $x\in[-1,1]$ and $y\in[0,2]$. My approach: Since, $|y-x^2|$ is positive ...
0
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1answer
35 views

How this integral is evaluated $\frac{\partial }{\partial x}\left(\int _y^x\cos \left(-5t^2-2t-4\right)\:dt\right)$?

How this integral is evaluated? $$\frac{\partial }{\partial y}\left(\int _y^x\cos \left(-5t^2-2t-4\right)\:dt\right)$$ And in general, are there general methods for partial differentiation ...
0
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1answer
29 views

Definite integral of trigonometric functions with complicated arguments

I came across beautiful integral (where $n$ is integer) $I(n, z) = \int_0^{\pi} \cos(nx) \sin(z \cos(x) ) \mathrm{d}x $ According to Gradshteyn and Ryzhik (p 414, Sec. 3.715, Eq. 13), solution is ...
2
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2answers
67 views

How much velocity can a canister of fuel give a spaceship?

I've recently considered the issue of how much velocity a canister of fuel can provide a 'spaceship'. I assumed we could approximate a basic solution If we know the mass of the fuel $m$, the mass of ...
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0answers
30 views

How to integrate discrete data by Gaussian quadrature method

I'm trying to numerically integrate discrete data by Gaussian quadrature method. The file attached test.mat is a discrete data set taken from a finite-element mode ...
1
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2answers
86 views

How to understand this integral result?

I was reading this page on Wikipedia: Birthday Attack I can understand up until how to approximate the minimal number of attempts for a given probability $$n(p; H) \approx \sqrt{2H \log \frac 1{1-p}...
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2answers
146 views

Evaluating $\int_0^{\infty} {\frac{\sin{x}\sin{2x}\sin{3x}\cdots\sin{nx}\sin{n^2x}}{x^{n+1}}}\ dx$

How to calculate $$ \int_{0}^{\infty}{\sin\left(x\right)\sin\left(2x\right)\sin\left(3x\right)\ldots \sin\left(nx\right)\sin\left(n^{2}x\right) \over x^{n + 1}}\,\mathrm{d}x $$ I believe that we ...
4
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2answers
115 views

Compute $\int\limits_Q^1\sqrt{(1-x^2)(1-\frac{Q^2}{x^2})}\mathrm{d}x$

I am trying to compute the integral $$\int\limits_Q^1\sqrt{(1-x^2)(1-\frac{Q^2}{x^2})}\mathrm{d}x$$ where $0\leq Q <1$ is a real number. I tried to substitute $x=\cos y,$ but this didn't bring ...
10
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3answers
231 views

How to prove that$\int_{0}^{1}\ln{(x/(1-x))}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$

$$\int_{0}^{1}\ln{\big(\frac{x}{1-x}\big)}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$$ Put $$\frac{x}{1-x}=y$$ $$I=\int_{0}^{\infty}\ln{y}\ln{(1+3y+y^2)}\frac{dy}{y(y+1)}=\frac{8}{5}\zeta{(3)...
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3answers
66 views

Choosing integration strategy [duplicate]

I need to integrate this function: $$\int \frac {x^3}{\sqrt{4-x^2}} \;dx$$ I don't know how i should integrate, should i divide because the numerator has an higher grade? There is any strategy i ...
2
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1answer
98 views

hard integral problems to solve

I'm practicing harder integration using techniques of solving with special functions I have difficulties with these two hard integrals; don't even know how to start, $$\int_0 ^\infty x^p e^{-\frac{\...
3
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0answers
33 views

Maximizing $\frac{\int_r^1xf(x)dx}{2-F(r)}$

Consider a continuous distribution on $(0,1)$ with probability distribution function $f$ and cumulative distribution function $F$. Define $$g(r)=\frac{\int_r^1xf(x)dx}{2-F(r)}$$ and let $r_M\in(0,1)$ ...
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10answers
2k views

What does it mean when dx is put on the start in an integral? [duplicate]

I have seen something like this before: $\int \frac{dx}{(e+1)^2}$. This is apparently another way to write $\int \frac{1}{(e+1)^2}dx$. However, considering this statement: $\int\frac{du}{(u-1)u^2} = \...
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1answer
22 views

Solving quasilinear PDE - 1D, time-dependant, convection

I have a task to solve the following quasilinear PDE (find $c(x,t)$): $$ c_x v + c_t = - v_x c $$ $c \in (0,20) , t \in (0, \infty)$ where I know function $v(x)$ to be $v(x) = \frac{3}{40}(1+\cos(\...
1
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1answer
407 views

limits of an integrable function over increasing sequence of sets

Let $(E_n)_{n \geq 1}$ be an increasing sequence of sets such that $\bigcup_{n \geq 1} E_n = \Omega$. Then for every integrable function $f$ we have $$\lim_{n \rightarrow \infty} \int_{E_n} f d\mu = \...
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1answer
33 views

Evaluate $\iiint_V zdV$, V is volume bounded below by cone $x^2+y^2 = z^2$ and above by sphere$ x^2+y^2+z^2=1$,lying on positive side of y-axis.

$\iiint_V zdV = \iiint_{\sqrt{x^2+y^2}}^{\sqrt{1-x^2-y^2}} zdzdydx = \iint_D \frac{1-2(x^2+y^2)}{2}dxdy$ where D is given by the disc $x^2+y^2 = \frac{1}{2}$ Changing x,y into cylindrical coordinates,...
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24 views

Integral over domain with implicit boundary

I have a unit disc (circle) $$x^2 + y^2 = 1$$ and a function $$\sum_{i=0}^N c_i\left(\sqrt{(x-a_i)^2 + (y-b_i)^2}\right)^3 = 0$$ I want to compute an area bounded by these two functions for $x$ ...
22
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3answers
1k views

Does integration by parts with “deja vu” have a name?

In some integration by parts problems, such as evaluating the integral of $e^x \cos x$ or $\sec^ 3 x$, one performs integration by parts (possibly more than once, and possibly together with algebraic ...
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1answer
15 views

Infinities on null sets

This is a conceptual question! Why is it that (e.g.) $\int_0^1 \frac{1}{x} dx$ doesn't converge. I'm stuck in the following way of thinking about it: Since the problematic part is $\int_0^\epsilon \...
4
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2answers
87 views

How can I solve this triple integral $\iiint_{B} y\;dxdydz$ on a defined set?

Calculate $$\iiint_{B} y\;dxdydz.$$ The set is $\;B=\{(x,y,z) \in \mathbb R^3$; $\; x^2+y^2+4z^2\le12$, $-x^2+y^2+4z^2\le6$, $y\ge 0 \}$. I know that B is defined by a real ellipsoid, an ...
0
votes
0answers
17 views

Volume of the $N$-dimensional domain $\sum\limits_{k=1}^N (1 + |x_k|^a)^b\le\epsilon$

I wish to calculate the following $N$-dimensional integral $$I = \int_0^\infty dx_1 \ldots \int_0^\infty dx_{N} \, H\left(\epsilon - \sum_{k=1}^N (1 + x_k^a)^b\right),$$ where $a, b$ and $\epsilon$ ...
0
votes
1answer
27 views

Integral substitution

I don't understand why the integral boundary change here from $[0,1]$ to $[0,\infty]$ $$\int_0^1 \int_{0}^\infty xe^{-x}f(ux,(1-u)x)\mathrm{d}u \mathrm{d}x$$ Substitution: $(ux=t,\ (1-u)x=s)\implies ...
3
votes
1answer
23 views

Converting Ellipse Integration Boundaries To Cylindrical Coordinates

I'm having the following integral, and I'm being asked to convert the integration boundaries to cylindrical coordinates. I've figured out that on XY-plane it's an ellipse having the following ...
11
votes
3answers
456 views

Meaning of $\int\mathop{}\!\mathrm{d}^4x$

What the following formula mean? $$\int\mathop{}\!\mathrm{d}^4x$$ I know that this $\int f(x)\mathop{}\!\mathrm{d}x$ is the integral of the function $f$ over the $x$ variable, but the following $\...
1
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1answer
71 views

Is there a reduction formula for $I_n=\displaystyle\int_{0}^{n\pi}\frac{\sin x}{1+x}\,dx$?

I haven't been able to manipulate this integral. I need to find the value of $I_n$ for $n=1,2,3,4$ and arrange them in ascending order.
0
votes
0answers
13 views

Correlation of an integral

Assuming that $a(x)$ is a random variable and correlation $\xi(x_1,x_2)=\langle a(x_1)a(x_2)\rangle$ is known (where angular bracket denote statistical averaging), it it possible to write the ...
1
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4answers
110 views

Find $\int_0^{\pi}\sin^2x\cos^4x\hspace{1mm}dx$

Find $\int_0^{\pi}\sin^2x\cos^4x\hspace{1mm}dx$ $ $ This appears to be an easy problem, but it is consuming a lot of time, I am wondering if an easy way is possible. WHAT I DID : Wrote this as $\...