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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0
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0answers
40 views

Rewrite order of $\int_0^1\int_0^{\sqrt{y}} \int_y^1 \, dz \, dx \, dy$ to $dx\,dy\,dz$ and $dy\,dz\,dx$

I need to change the order of $$\int_0^1\int_0^{\sqrt{y}}\int_y^1\,dz\,dx\,dy$$ to $dx\,dy\,dz$ and $dy\,dz\,dx.$ I can extract the inequalities to get $1≤z≤y$, $0≤x≤\sqrt y$, $0≤y≤1$, but I get ...
2
votes
1answer
54 views

$\int \sqrt{4-\tan^2x} \sec^2x \, dx $ via substitution method

I am trying to determine via substitution method $$\int \sqrt{4-\tan^2x} \sec^2x \, dx$$ $$t = \tan x $$ $$dt=\sec^2x\,dx$$ $$\int \sqrt{4- t^2} \, dt$$ $$t=2\cos\theta$$ $$dt=- 2\sin\theta \, d\...
5
votes
2answers
445 views

Find the average value of the function…

Find the average value of the function $$F(x) = \int_x^1 \sin(t^2) \, dt$$ on $[0,1]$. I know the average value of a function $f(x)$ on $[a,b]$ is $f_\text{avg}=\dfrac1{b-a} \int_a^b f(x) \, dx$, ...
5
votes
2answers
37 views

Assuming $\sum_{n = 1}^\infty \int |f_n| < \infty$, properties that follow for integral

How do I see that if $\sum_{n = 1}^\infty \int |f_n| < \infty$, then $\sum_{n = 1}^\infty f(x)$ converges absolutely almost everywhere, is integrable, and its integral is equal to $\sum_{n = 1}^\...
2
votes
0answers
38 views

why do we use dy/dx as ratio though it is not while solving the problems of integration by substitution [duplicate]

According to my knowledge dy/dx is not a ratio. Then while solving the problems of integration by substitution how can we use it as ratio. Because of we have dx/dt =f(x). Then while shoving it by ...
0
votes
1answer
58 views

Integrate function by partial derivative

I'm searching a $\phi(x,t)$ solution of a pde cauchy system, with $x\in[-1,1],t\in[0,T]$ I am able to know: a) $\phi(x,0)=-cos\left(\pi\left(x-0.85\right)\right)$ b) $\phi_x(x,t)$, $\forall t,x$ (...
1
vote
3answers
42 views

Area between four functions

I'm not sure about calculating the area between: $f(x)=x+3$ ,$g(x)=x^2-9$,$k(x)=5$ and $y=0$ My idea , but i'm not sure about the first integral, is: $$$$ $\int_\sqrt{14}^05-(x-3)dx$ + $\int_0^25-(...
2
votes
1answer
51 views

Deriving an Expression for the Coordinates of a Partial Hollow Torus as a Function of the Angle

I'm modeling a shape that is best described as a partial, hollow torus. Here's what it looks like: http://i.imgur.com/3h4H5KQ.png In my application, the angle can vary from 0 to 85 degrees. I'm ...
2
votes
9answers
239 views

Integrate $\int\frac{x+1}{\sqrt{1-x^2}} \; dx$ without using trigonometric substitution

I want to solve: $$\int\frac{x+1}{\sqrt{1-x^2}} \; dx$$ I know how to solve this using trigonometric substitution, but how can I solve the integral in an other way ?
0
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0answers
31 views

Simplify an Integration with cosine, logarithm and hyperbolic functions

I would like to simplify (or solve) the following integral: $$F(t):=\int_{a}^{t-a} \cos\left(\ln\left( \frac{f(t-x)}{f(x)} \right)\lambda \right)\frac{1}{\sqrt{f(t-x)\cdot f(x)}} dx, \quad t>2a,$$ ...
1
vote
1answer
33 views

Meaning of limits, $\int_{\max(0, t-5)}^{\max(0,t-3)} e^{-3s} \, ds $?

What does it mean to have $\max(0,t-3)$ and $\max(0,t-5)$ in the limits? Is it a abbreviation? $$ \int_{t-5}^{t-3} e^{-3s}u(s) \, ds = \int_{\max(0, t-5)}^{\max(0,t-3)} e^{-3s} \, ds $$ Source of ...
20
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3answers
459 views

The entry-level PhD integral: $\int_0^\infty\frac{\sin 3x\sin 4x\sin5x\cos6x}{x\sin^2 x\cosh x}\ dx$

I hope you find this integral interesting. Evaluate $$\int_0^\infty\frac{\sin\left(\,3x\,\right)\sin\left(\,4x\,\right) \sin\left(\,5x\,\right)\cos\left(\,6x\,\right)}{x\,\sin^{2}\left(\,x\,\...
1
vote
1answer
27 views

Changing the order of integration in three dimensions, of a level curve, $e^{x^2}$

Change the order of integration $dx\,dy$ to $dy\,dx$ and evaluate it: $$\int_0^1\int_{3y}^3 e^{x^2} \, dx \, dy$$ now I now that I can do the following: $$x=3,\quad x=3y \to y=\frac{1}{3}x$$ Okay ...
1
vote
3answers
74 views

How to integrate $\int \dfrac{1}{\sin^4 x \cos^4 x} dx$

The integral in question is: $$\int \dfrac{1}{\sin^4 x \cos^4 x} dx$$ I tried using $1 = \sin^2 x + \cos^2 x$, but it takes me nowhere. Another try was converting it into $\sec$ and $\csc$, but ...
1
vote
1answer
44 views

Finding $\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $

As the question says, $$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $$ where a is a constant, $a>0$.
0
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4answers
52 views

Area of circle (double integral and cartesian coordinates)?

I know that the area of a circle, $x^2+y^2=a^2$, in cylindrical coordinates is $$ \int\limits_{0}^{2\pi} \int\limits_{0}^{a} r \, dr \, d\theta = \pi a^2 $$ But how can find the same result with a ...
0
votes
1answer
36 views

$\int\frac{x }{ \sqrt {3-x^4}}{dx} $ by substitution method

I am trying to determine via substitution: $$\int\frac{x }{ \sqrt {3-x^4}}{dx} $$ My work: $$x=\frac{1}{t}$$ $$dx=-\frac{dt}{t^2}$$ $$ - \int\frac{dt }{ t \sqrt {3t^4-1}} $$ How to proceed ...
21
votes
6answers
1k views

Conjectured value of $\int_{0}^{\infty}\left(\frac{x-1}{\ln^2 x}-\frac{1}{\ln x}\right)\frac{\mathrm{d}x}{x^2+1}$

I was curious whether this integral has a closed form expression : $$\int_{0}^{\infty}\left(\frac{x-1}{\ln^2 x}-\frac{1}{\ln x}\right)\frac{\mathrm{d}x}{x^2+1}$$ The integrand has a singularity ...
2
votes
3answers
66 views

how to partial fraction $\frac{1}{(x+1)^2}$

I need to integrate $\frac{1}{(x^2+2x+1)}$, so I need to use partial fraction as the polynomial can be factored as $\frac{1}{(x+1)^2}$. This is what I've tried: $$\frac{A}{(x+1)} + \frac{B}{(x+1)^2}$$...
4
votes
3answers
56 views

integrate $\int \frac{x^4-16}{x^3+4x^2+8x}dx$

$$\int \frac{x^4-16}{x^3+4x^2+8x}dx$$ So I first started with be dividing $p(x)$ with $q(x)$ and got: $$\int x-4+\frac{8x^2+32x-16}{x^3+4x^2+8x}dx=\frac{x^2}{2}-4x+\int \frac{8x^2+32x-16}{x^3+4x^2+...
1
vote
1answer
41 views

Find $f$ such that $\int^b_a f^2(x) = c$ and $\int^b_a f(x)g(x) dx$ should be maximum

Given $g$ integrable on $[a,b]$, find integrable $f$ such that $\int^b_a f^2(x) = c > 0$ and $\int^b_a f(x)g(x) dx$ should be maximum. I tried to use Cauchy–Schwarz inequality and define $d:= \...
2
votes
2answers
45 views

Calculation double Integral over Ball (optical size)

I hope that someone can help me with the following problem. I have to show that $$\int_{B_1(0)}\int_{B_1(0)}\frac{1}{|x-y|^2}dxdy=4\pi^2~,$$ with $B_1(0)\subset\mathbb{R}^3$. I have no idea how to ...
1
vote
3answers
96 views

Evaluation of $4\int_0^{+\infty} \frac{\left(\sinh\left(\frac{x}{8}\right)\right)^2}{x(e^x-1)}dx$

In relation with Evaluating series of zeta values like $\sum_{k=1}^{\infty} \frac{\zeta(2k)}{k16^{k}}=\ln(\pi)-\frac{3}{2}\ln(2) $ From the well-known formula, For $s$, such that $\Re(s)>1$, $\...
-3
votes
1answer
61 views

Calculus 3 Integrals

Find the volume of the solid bounded by $z=4x^2+4y^2, z=0, x^2+y^2=1$ and $x^2+y^2=2$. What I know: I know that when I draw the graph I will get two paraboloids giving me a radius of $1$ and $2$,$\...
1
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0answers
46 views

Isi 2016 problem no. 7 integration [duplicate]

$f$ is a differentiable function such that $f(f(x))=x$ where $x \in [0,1]$.Also $f(0)=1$.Find the value of $$\int_0^1(x-f(x))^{2016}dx$$
0
votes
2answers
36 views

Triple integral using Cylindrical Coordinates

Evaluate the iterated integral $x= 0$ to $x=1$, $y= -\sqrt{1-x^2}$ to $y=\sqrt{1-x^2}$ and $z= 0$ to $z=2-x^2-y^2$ and $f(x,y,z)= \sqrt{x^2+y^2}$ What I know: I know that I have to use cylindrical ...
1
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0answers
27 views

Simplifying Multiple Integral for Compound Probability Density Function

Are there any ways to simplify this multiple integral? $$ \hat{f}\left(\left.y\right|\alpha\right)=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\hat{f}\left(\left.y\right|\theta_{1}\right)\hat{...
3
votes
2answers
47 views

Integrating over a shell in $d$-dimensional space.

In M. E. Tuckermann's book Statistical Mechanics: Theory and Molecular Simulation, par. 3.2, the following approximation is made: $$\int_{c<f(\vec x)< c+\epsilon} d^D x \simeq \epsilon \int_{\...
0
votes
0answers
50 views

Integral Calculus with non integer power using Cauchy integral's theorem or Residue theorem

I want to calculate the following integral: $$I(\beta)=\int_0^\infty \frac{\sin(x)}{ x^{\beta+1}}dx$$ for $ 0 \leqslant\beta<1$ I used comparison test for improper integrals to be sure that these ...
0
votes
0answers
12 views

Integration with lower order error terms affecting integral

I want to upper bound the the following integral: $$ \int_{R/2}^R e^{kx} \cdot (1 + \mathcal O(e^{R-2x}))^{k} \,\mathrm d x, $$ where $k < e^{R/2}$, and $R$ is large. Now by just observing the ...
2
votes
1answer
48 views

Integral involving Dirac delta: two different results?

I am evaluating the integral over all space $$\int \delta \left(r^2 - R^2\right) d \vec r$$ At first, I did this: $$\int \delta \left(r^2 - R^2\right) d \vec r = 4 \pi \int_0^\infty \delta \left(r^...
1
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0answers
37 views

Is an average function integrable?

I'm thinking about the following question: If $u\in L^p(\mathbb{R}^n)$, is $f(x)=\int_{|y-x|<R}|u(x)-u(y)|^pdy$ in $L^1(\mathbb{R}^n)$, where $R>0$ is a fixed numbers? It's clear that if $u$ ...
-1
votes
0answers
21 views

Change the order of integral

How to change the integral $\int_a^y G(x)\int_a^x F(t) \int_a^t W(z) dzdtdx$ into the form $\int_{?}^{?} F(x) \int_a^{?} G(t) \int_a^{?} W(z)dzdtdx$
0
votes
0answers
11 views

Integral after applying monotonic function

Let $F$, $G$ are two cdf contained in [a,b], $w$ is a non-negative and strictly increasing probability weighting function. If $\int_a^b G(x)-F(x) dx =0$, and $\int_a^y P(x)\int_a^x G(z)-F(z)dzdx \geq ...
1
vote
1answer
35 views

Resistance of a frustum, or a pyramid cut from top

We know that Resistance is given by: $$R = \dfrac{\rho L}{A}$$ The figure of Resistance is like this, $L$ is perpendicular distance between $A_1$ and $A_2$ I assumed that the area change linearly, ...
3
votes
1answer
32 views

Alternate proof of the dominated convergence theorem by applying Fatou's lemma to $2g - |f_n - f|$?

Here is a proof of the dominated convergence theorem. Theorem. Suppose that $f_n$ are measurable real-valued functions and $f_n(x) \to f(x)$ for each $x$. Suppose there exists a nonnegative ...
7
votes
1answer
37 views

Variant of dominated convergence theorem, does it follow that $\int f_n \to \int f$?

Suppose $f_n$, $g_n$, $f$ and $g$ are integrable, $f_n \to f$ almost everywhere, $g_n \to g$ almost everywhere, $|f_n| \le g_n$ for each $n$, and $\int g_n \to \int g$. Does it follow that $\int f_n \...
4
votes
1answer
27 views

$\sup_n \int |f_n|^{1 + \gamma}d\mu < \infty$ implies $\{f_n\}$ is uniformly integrable?

Suppose $\mu$ is a finite measure and for some $\gamma > 0$, we have$$\sup_n \int |f_n|^{1 + \gamma}d\mu < \infty.$$Does it follow that $\{f_n\}$ is uniformly integrable?
3
votes
4answers
171 views

How solve $\int_{0}^{\infty} \dfrac{1-\cos x}{x^{2}} dx$ [closed]

What is the value of the following integral? $$\int_{0}^{\infty} \dfrac{1-\cos x}{x^{2}} dx$$
0
votes
0answers
36 views

Proving that an infinite series equal a finite series

Suppose we have a function $f(z)$, which has $m$ isolated singularities, which are non-integers (say, $z_1$, $z_2$,...,$z_m$). Define $H(z):=\frac{\pi f(z)}{\sin(\pi z)}$. Assume that there exists a ...
3
votes
1answer
25 views

Does it follow that $\{f_n\}$ is uniformly integrable?

Suppose $\mu$ is a finite measure, $f_n \to f$ almost everywhere, each $f_n$ is integrable, $f$ is integrable, and $\int |f_n - f| \to 0$. Does it follow that $\{f_n\}$ is uniformly integrable?
0
votes
2answers
34 views

Integral of the level curve $ye^{2x}$

$\int_{0}^{2}\int_{0}^{4} ye^{2x}$ $dydx$ The book says the answer is $32(e^4-1)$ I had to do u-substitution when doing it, maybe that's where I went wrong. First integrating with respect to y, ...
1
vote
1answer
64 views

How to prove that : $\int_{2}^{x}\frac{dO(te^{-c\sqrt{\log t}})}{t\log t}=O(1)$ [closed]

$\displaystyle{ \mbox{How I can prove that}\ \int_{2}^{x}{\,\mathrm{d}\alpha\left(t\right) \over t\,\log\left(t\right)} = \,\mathrm{O}\left(1\right)\ ?,}\quad $ where $\alpha\left(t\right) = \,\mathrm{...
4
votes
3answers
119 views

What is the value of $I=\lim_{n \to \infty} \int_0^1 {{1 + nx^2}\over{(1 + x^2)^n}} \log(2 + \cos(x/n))\,dx.$?

Find the integral $I$.....it looks like a good problem which I was not able to solve ....please help... $$I=\lim_{n \to \infty} \int_0^1 {{1 + nx^2}\over{(1 + x^2)^n}} \log(2 + \cos(x/n))\,dx.$$
6
votes
1answer
71 views

Do we necessarily have that $\int g\,d\mu_n \to \int_0^1 g\,dx$?

Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0, 1]$. Suppose $\mu_n$ are finite measures on $([0, 1], \mathcal{B})$ such that $\int f\,d\mu_n \to \int_0^1 f\,dx$ whenever $f$ is a real-valued ...
8
votes
2answers
50 views

$\{f_n\}$ is uniformly integrable if and only if $\sup_n \int |f_n|\,d\mu < \infty$ and $\{f_n\}$ is uniformly absolutely continuous?

Let $(X, \mathcal{A}, \mu)$ be a measure space. A family of measurable functions $\{f_n\}$ is uniformly integrable if given $\epsilon$ there exists $M$ such that$$\int_{\{x : |f_n(x)| > M\}} |f_n(x)...
4
votes
1answer
27 views

Countable collection of Borel subsets of $[0, 1]$, exists subsequence where $\int_A f_{n_j}(x)\,dx$ converges for each $i$?

Let $\{f_n\}$ be a sequence of measurable real-valued functions on $[0, 1]$ that is uniformly bounded. How do I see that if $\{A_i\}$ is a countable collection of Borel subsets of $[0, 1]$, then there ...
1
vote
3answers
37 views

How do I see that if $A$ is a Borel subset of $[0, 1]$, then there exists a subsequence $n_j$ such that $\int_A f_{n_j}(x)\,dx$ converges?

Let $\{f_n\}$ be a sequence of measurable real-valued functions on $[0, 1]$ that is uniformly bounded. How do I see that if $A$ is a Borel subset of $[0, 1]$, then there exists a subsequence $n_j$ ...
1
vote
1answer
17 views

Applying the Cartesian Coordinate

Find the surface area of the portion of $2x+y+z=8$ in the first octant. I know that $f(x,y)=8-y-2x$, $x=0$ to $4$ and $y= 0$ to $8-2x$, but I'm having trouble solving the problem in Cartesian form. ...
2
votes
1answer
21 views

seq. of nonneg. Lebesgue measurable functions on $\mathbb{R}$, have $\limsup_{n \to \infty} \int f_n\,dx \le \int \limsup_{n \to \infty} f_n\,dx$?

Let $f_n$ be a sequence of nonnegative Lebesgue measurable functions on $\mathbb{R}$. Is it necessarily true that$$\limsup_{n \to \infty} \int f_n\,dx \le \int \limsup_{n \to \infty} f_n\,dx?$$If not, ...