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 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
13 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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0answers
15 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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2answers
21 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) ...
3
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1answer
46 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 ...
2
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2answers
46 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
11 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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1answer
72 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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2answers
20 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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0answers
21 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 ...
3
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0answers
27 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 ...
0
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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
26 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
45 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
21 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} ...
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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0answers
11 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
11 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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1answer
31 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
26 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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0answers
35 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 ...
0
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0answers
26 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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0answers
24 views

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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0answers
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?
4
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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: ...
2
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1answer
19 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
16 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 ...
3
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0answers
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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0answers
21 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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0answers
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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0answers
26 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
81 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 ...
3
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3answers
46 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
32 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 ...
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0answers
94 views

Integral Contest [on hold]

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 ...
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1answer
38 views

How to calculate this kind of integral? [on hold]

General form: $$ \int \sqrt{\alpha x^3 + \beta x^2 + \gamma x + \delta} \, dx $$ Example: $$ \int \sqrt{\frac{2}{3} x^3 + x^2 + 4} \, dx $$ Please describe more details.(I'm a freshman.) I will ...
7
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0answers
52 views

Real analytic methods for the following integral

A few days back, the following integral was posted $$\int_0^1 x^x(1-x)^{1-x}\sin(\pi x)\,dx=\frac{\pi e}{24}$$ The integral was answered using complex analysis tools but I am interested in other ...
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2answers
23 views

feedback on my solution (improper integral)

i have done this improper integral but i am not sure if i have followed the correct procedure or my answer is correct. Please help!
3
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1answer
28 views

feedback on my solution (integration)

I need help in this problem. I managed to find the answer for this problem by using mathmatica but cannot do the working for it. i have done most of it but i am stuck on the last part.
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0answers
32 views

how to calculate the following definate integral [on hold]

∫f(a,b,x)dx= Antiderivative or integral could not be found.
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0answers
35 views

A Riemann Integrability Question

Define $f:\mathbb{R} \rightarrow \mathbb{R}$. For any fixed closed interval $[a,b] $,$f(x) $ is $Riemann$ integrable on $[a,b].$ Proof:$\forall a,b;c,d\in\mathbb{R},a<b,c<d.$ $f (x+y) $ is ...
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0answers
35 views

Please solve this basic question [on hold]

integral of ${sec}^ \theta\, dx$ "i need the answer immediately"
2
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1answer
52 views

Really tricky integration-----double U and trig substitution

The definite integral $h(x) = \sin x/(1 + x^2)$ on the closed interval $[-1,1]$ $\tan^2(x) + 1 = \sec^2(x)$, $x = \tan(@)$ $$x = \tan(@)$$ $$dx = \sec^2(@) d@$$ now I have to find sin9x0 in terms ...
0
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0answers
12 views

integration featuring the unit step function

Compute the following integrals I don't know how to use MathJaX so here's a link to the image of the integrals where u(t) is the unit step function and σ is some variable of integration
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vote
2answers
41 views

How do I approach this double integral?

Let $R$ be the region inside $$x^2+y^2 = 1$$ but outside $$x^2+y^2 = 2y$$ with $x \ge 0 $ and $y \ge 0$ Let $$u = x^2 + y^2$$ and $$v = x^2+ y^2 - 2y$$ Compute $ \iint_R xe^y dxdy$ using this change ...
0
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0answers
16 views

Integration by parts partial derivatives

Given $$\int_x \int_t \Big( \frac{\partial}{\partial t}u(x,t) + \frac{\partial}{\partial x}f(u(x,t)) \Big) \phi(x,t)~~ dt dx = 0$$ How can I apply integration by parts in order to have the ...