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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8 views

Solving the integral which shows the second moment of subtracting two Beta-distributed Random Variables

Peace be upon you In my project I needed to find the second moment of the subtraction of two Beta-distributed random variables. I have computed it and reached to the following integral which I should ...
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1answer
12 views

Fubini's theorem, interchanging order of integration

My question is, imagine I want to compute the following integral: $$\int_A \int_B f(x,y)dxdy$$ and I decide to start from $x$ and get $$\int_A \int_B f(x,y) dxdy <\infty.$$ On the contrary if I ...
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12 views

Calculating the Integral of a non conservative vector field

I have no clue how to do part C because a) is non conservative What I got for b) $f(x,y)=\frac{x^3}{3}+2yx+\frac{y^3}{3}+K$ (I don't know the symbol for the thing so I used f(x,y) instead. How do I ...
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0answers
8 views

Estimates for the wave equation

Spose $ u $ solves the wave equation on $ U \subset \mathbb{R}^3 $ with initial conditions $ u (x, 0) = g(x)$ and $ u_t(x,0) = h(x)$, where lower script indicates partial differentiation. Then we have ...
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2answers
20 views

Integrating the gamma function

I assumed that $$\Gamma\left(k+\frac{1}{2}\right)=2\int^\infty_0 e^{-x^2}x^{2k}\,dx=\frac{\sqrt{\pi}(2k)!}{4^k k!} \,,\space k>-\frac{1}{2}$$ and that ...
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30 views

How do I do this change of variables?

Use a change of variables to evaluate: $$\iiint\limits_{D}xy\,\mathrm{d}V$$$D$ is bounded by the planes $y-x=0$, $y-x = 2$, $z-y = 0$, $z-y = 1$, $z=0$, $z=3$. I set $$u = y-x$$ $$v = z-y$$ $$w ...
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1answer
19 views

A suitable integration path for $\cos z/(2 + \cos z)$

I was solving the exercises and got stuck when trying to solve this with tools of residual calculus $$ \int_{0}^{2 \pi} \frac{\cos (z)}{2 + \cos (z)} \, dz = \int_{0}f(z) \, dz. $$ Isolated ...
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1answer
27 views

Calculus: volume of revolution about a line other than the $x$-axis.

Find the volume of the solid of revolution obtained by rotating the region bounded by $f(x) = x^3 + 1$, $g(x) = x^2$ and $0 ≤ x ≤ 1$ about the line $y = 3$. I know the gist of the problem, but ...
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2answers
22 views

Calculate line integral

For $f(x,y) = 2x + y + 10$, calculate the line integral $$ \int_{L}{f(x,y)dL} $$ where $L$ is the straight line between $(1,4)$ and $(5,1)$ in the $xy$-plane.
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3answers
69 views

Integration $I_n=\int_{0}^{1}\frac{dx}{(x^n+1)(\sqrt[n]{x^n+1})}$

$$I_n=\int_{0}^{1}\frac{dx}{(x^n+1)\large\sqrt[n]{\normalsize x^n+1}}$$ Could someone help me through this problem?
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1answer
38 views

Integration by Substitution

Question: Use the substitution $u=\tan x$ to find $\displaystyle \int_{0}^{\large \frac{\pi}{4}}\left(\tan^{n+2}x + \tan^{n}x\right)dx$. Using the above result, find the exact value of ...
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0answers
9 views

How to relate two integration contour?

How one can relate two integration contour? For example if I have an integration contour like $\int_{-a}^{a}f(x)dx$ here let say a=infinity. How I can say that the integral $\int_{2}^{3}f(x)dx$ is a ...
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1answer
18 views

Determine integral by using the following identity (which is imaginairy)?

I want to determine the following integral: $$\int_{-\infty}^\infty \frac1{x^6+1} dx$$ by using the following identity: $$\frac1{x^6+1} = \Im\left[\frac1{x^3-i}\right]$$ How in the world can I do ...
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2answers
16 views

Integration separation of variable

Liquid is flowing into a small tank which has a leak. Initially the tank is empty and, t minutes later, the volume of liquid in the tank is V cm3 . The liquid is flowing into the tank at a constant ...
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1answer
45 views

Closed-form of $\int_0^1 \int_0^1 \int_0^1 x^{(y^z)} \,dz\,dy\,dx$

We know that $$\int_0^1 \int_0^1 x^y\,dy\,dx = \ln 2.$$ Do we know a closed-form of $$\int_0^1 \int_0^1 \int_0^1 x^{(y^z)} \,dz\,dy\,dx\,?$$ As a start we know that $$\int_0^1 x^{(y^z)}\,dz = ...
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26 views

By applying the second version of the Fundamental Theorem of Calculus find the integral:

The second version of the Fundamental Theorem of Calculus states that if $F'(x)=f(x)$ then $\int_{a}^{b} f(x) dx = F(b)-F(a)$. I need to use this to find a) $\int_{-2}^{-1} \frac{1}{x^3} dx $ and b) ...
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1answer
49 views

Integrate $e^{-\frac{y^2}{2}}\left(\frac{1}{y^2}+1\right)$

I'm trying to find $$\displaystyle \int{e^{-\frac{y^2}{2}}} \left(\frac{1}{y^2}+1\right)dy$$ I tried using integration by parts and some substitutions, but nothing seem to work. The answer is ...
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2answers
52 views

Evaluate $\int\sec^4(u) \operatorname d \!u$

Evaluate $$\int\sec^4(u) \operatorname d \!u$$ I don't know what to substitute: I've tried $1+\tan(u)$ and integration by parts. I know the general formula for $\sec^n(u)$, but I want to be able to ...
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0answers
14 views

Setting up a volume-finding calculation

I'm asked to find the volume inside the sphere $x^2+y^2+z^2=25$ and outside the cylinder $x^2+y^2=1$. I approached the volume $V$ in the following way: ...
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2answers
115 views

How to evaluate $\int_0^1\frac{\tanh ^{-1}(x)\log(x)}{(1-x) x (x+1)} \operatorname d \!x$?

How to evaluate the following integral $$\int_0^1\frac{\tanh ^{-1}(x)\log(x)}{(1-x) x (x+1)} \operatorname d \!x $$ The numerical result is $= -1.38104$ and when I look at it, I have no idea how to ...
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23 views

how can i change specifically the intervals of a double integral?

I know how to change the intervals of an integral, for example the integral of $(\sin x)^2$ from $-\pi$ to $\pi$ is equal to $\pi\int_{-1}^1 (\sin πx)^2 \,dx$. I find it difficult to do that in 2D. ...
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23 views

Is there a way to use this interpretation of differential forms on manifolds?

I read Rudin's "Principles of Mathematical Analysis". In the part of Differential Forms, he defined them formally. I particularly enjoyed the formal viewpoint, since everywhere else it seems that the ...
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0answers
30 views

How to find the Maclaurin series for the integral of $e^{x^2}$?

I am trying to find the Maclaurin series for the integral of $e^{x^2}$? What I done so far is that the Maclaurin series for $e^{x^2}$ is $$e^{x^2}=\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}$$ So would ...
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1answer
21 views

$f, g: X \rightarrow \bar{R}$ are measurable, if $f \leq g$ a.e. then $\int f d\mu \leq \int g d\mu$

Let $(X,M,\mu)$ be a measure space. $f, g: X \rightarrow \bar{R}$ are measurable. If $f \leq g$ a.e. and $\int f d\mu, \int g d\mu$ both exist, show that $\int f d\mu \leq \int g d\mu$. Here a.e. ...
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2answers
28 views

Convergence of sequence of integrals.

Let $(\mathcal{X}, \mathcal{A}, \mu)$ be a measure space, $f_n: \mathcal{X} \to \Bbb R$ a sequence of measurable functions, and $g_n:\mathcal{X} \to \Bbb R$ integrable functions such that $|f_n| ...
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2answers
46 views

Any idea ? $\int {\sqrt{1+\sqrt{x}}}/x dx$

$\int \frac{{\sqrt{1+\sqrt{x}}}}{x} dx$ I try with $u=\sqrt x $ but i don't know what to do... Thank you Shadock
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1answer
22 views

how to show this manipulation in the integral

Let we have: $$G(t)=y_1(t)\int y_2(s)ds$$ when we take the limits as: $$G(t)=y_1(t)\int^t_{t_0} y_2(s)ds$$ then is it possible to write it as: $$G(t)=y_1(t)\int^t_{t_0} y_2(s)ds=\int^t_{t_0} ...
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1answer
23 views

Center of mass of a barrel partially filled with grain

Bubba has a barrel in the shape of a cylinder of mass 39.4 kg. The barrel has a diameter of 62.4 cm and is 1.32 m tall. He fills the barrel to a depth of 49.5 cm with loose packed grain that has an ...
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15 views

Triple integral containing definite integral and exponentials with trigonometric functions

I am attempting to solve the following integral analytically: $$ \int_{z=5i}^{z=1} \int_{t=\csc^{-12}(z)}^{t=2} \int_{\theta=\sin^{t}(z)}^{\theta=t^2} {[\mathrm{e}^{t\cos(\mathrm{e}^{i \theta})} + ...
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3answers
13 views

Integrate the differential equation of a simple rate equation

Could somebody please show me how to integrate the following: $dA/dt = -kA$ I'm told that the answer is: $A(t) = A(0)e^-kt$ but I do not know why. Could you be explicit in your answer and explain ...
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1answer
26 views

Calculating a limit of integrals

I am having a problem with the following exercise: Show that for every bounded borelian function $\varphi : \mathbb{R} \rightarrow \mathbb{R}$, $\underset{n}{lim} \frac{n}{\sqrt{2\pi}} ...
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35 views

Equivalent form of a double integral.

I am looking at the second question of this problem set: The iterated integral $\int_0^1 \int_{y/2}^1 e^{x^2} dx \, dy$ can be expressed as (a) $\int_0^1 \int_0^{2x} e^{x^2} dy \, dx$ ...
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2answers
27 views

Fundamental theorem of calculus problem - trig functions

My problem is: On the interval (0 , pi/2). I know I need to split it in two integrals, but I don't know how. I would appreciate any suggestions on how to proceed.
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2answers
37 views

Integral with parameter: $\int_{0}^{a}x^2\sqrt{a^2-x^2}dx$

I have the following integral : $$\int_{0}^{a}x^2\sqrt{a^2-x^2}dx$$ I tried to manipulate the integral and then use substitution to get a rational form to arrive at: $$-8a^4\int_0^a ...
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36 views

Integration question with square root

$\int\sqrt {(1+3x^2+6x^3}dx$ I tried taking the substitution $u^2=1+3x^2+6x^3$. I was able to simplify the integral to $\frac{1}{(3x)(1+3x)} + \frac{x}{1+3x} + \frac{2x^2}{1+3x}$. I know I can form ...
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27 views

Proving $\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}{e^{\cos(x)}\cot(x)} dx < \frac{1}{e}$

While i was playing around with very weird functions and came across this: $$ \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}{e^{\cos(x)}\cot(x)}dx \approx 0.3676932086...\approx \frac{1}{e} - ...
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Does there exist such function?

Fix an integer value $k\geq 1$. Let $[0,1]$ the unit interval and let $s\in [0,1]$. Does there exist a function $f$ (which depends on $k$ of course but not on $s$) such that $$\int_s^1 \left( ...
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20 views

Integrate $x^{7/3}e^{-x}$ from $0$ to $\infty$

Yes, there's the method of converting this function to gamma function of 10/3. Is there any other way?
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3answers
37 views

Calculating a limit with infinitely many terms

I've encountered this limit : $$\lim_{n\to\infty} \frac1n \left(\sin\left(\frac{\pi}{n}\right) + \sin\left(\frac{2\pi}{n}\right)+\cdots+\sin{\frac{(n-1)\pi}{n}}\right)$$ Wolfram gives the value: ...
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1answer
20 views

Improper integral: is it convergent?

Is this integral finite? $$\int_s^t \frac{dx}{x^{1/2} - s^{1/2}}$$ where $s,t \in (0,\infty)$. More generally, I have the following integral $$\int_s^t ...
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0answers
17 views

$\int_{\Omega}a(t)\, d\mu(t)=\text{const}\Rightarrow a\equiv\text{const}$?

As the title already says I am wondering if $\int_{\Omega}a(t)\, d\mu(t)=\text{const}\Rightarrow a\equiv\text{const}$, whereat $\mu$ is a normalized measure on $\Omega$ Do not know exactly how ...
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27 views

Which integrals can be solved using Feynman's Technique?

How to check whether an integral can be easily solved using Feynman's approach. What are the main criteria needed to be taken into account?
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34 views

Multivariable integral over a simplex

Let $p$ be a positive integer, let $B > A >0$ and let $\beta >0 $ and $\beta \neq 1/2$. With a help of Mathematica (ie using elementary integration and consecutive simplifications) I have ...
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30 views

A complicated question

I have the following operator $A: H^1_{0,p}\longrightarrow H^1_{0,p}$ be defined by \begin{equation} Au(t)=\int_0^{+\infty} G(t,s)q(s)f(s,u(s))\,ds-\sum_{k=0}^{+\infty}G(t,t_k)h(t_k)I(u(t_k)), ...
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27 views

Convert time derivative to a function of time

Physics: I am asking for help to derive a general expression for the total amount of energy lost as a function of time from a radiating object. I'll simplify my problem like this: Say for example ...
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5answers
85 views

Find $\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$

How can we find the integral: $$\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$$ I tried to find and got it to be $\cfrac{\pi}{\sqrt2}$. Am I correct? Please help me with an ...
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3answers
47 views

Solve for $\int \sqrt{x}(\sqrt{x}-2x)^2 dx $

$\int \sqrt{x}(\sqrt{x}-2x)^2 dx $ so I solved this using U-substitution where $u= \sqrt{x}$ so my $du2\sqrt{x}=dx$ then it will be $2 \int u^2(u-2u^2)^2$ and just expand then distribute the $u^2$ so ...
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0answers
18 views

Question regarding double integrals

Regarding the Buffon's needle case for long needles of length $ l>t, $ (the distance between the parallel lines on the floor), we need to solve the integral $$ \int_{\theta=0}^{\frac{\pi}{2}} ...
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1answer
33 views

Evluating triple integrals via Spherical coordinates

Use Spherical coordinates to evaluate the triple integral $$\iiint_{\mathrm{x^2+y^2+z^2<z}}\sqrt{x^{2}+y^{2}+z^{2}}\, dV,$$ What I tried Converting $x^2+y^2+z^2<z$ to Spherical coordinates ...
4
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148 views

Integral Contest

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...