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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1answer
16 views

Change of integrals

This is probably ridiculously easy, but I can't figure this out: Why is $$\int_0^\infty \frac{e^{-t}}t \int_{|x-y|^2/(2t)}^\infty e^{-s} \,ds\,dt =\int_0^\infty e^{-s} \int_{|x-y|^2/(2s)}^\infty ...
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0answers
4 views

proof of a special case of discrete-time tower property

I'm reading a book on stochastic process and the first chapter is about properties of conditional expectation. One of the example the book gives is the proof of a special case of tower property in ...
0
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1answer
23 views

dominated convergence theorem and solve an integral

I will solve this integral from dominated convergence theorem can you show me how: $$\int^1_0\frac{t^{a-1}}{1+t^{b}}dt=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{nb+a}$$
2
votes
0answers
54 views

To determine a definite integral

I have been trying to solve the following integral $$\int_{0}^{\frac {\pi}{2}} \ln\left (\frac {\ln^2 (\sin x)}{\pi^2+\ln^2 (\sin x)}\right) \frac {\ln \cos x}{\tan x} dx$$ I tried substituting for ...
2
votes
2answers
24 views

When to use Integral Substitution?

$e^x$$(1+e^x)^{1\over{2}}$ why can't use integration by part, What is meant by in the form of f(g(x))g'(x)? Can you give a few example? Thank you
2
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1answer
32 views

Double integral polar coordinates for area bounded by curves

I need to find the area bounded by the curves : $$ x^2 + y^2 = 1, \ y^2= x\sqrt3, \ x \geq \frac{y^2}{\sqrt3} $$ My attempt : $$ \int^{\frac{\pi}{2}}_{\frac{3\pi}{2}} \int^{\text{some complicated ...
5
votes
2answers
43 views

How to evaluate $\int_0^{\infty} \bigg(\frac{e^{-x}}{\sinh(x)} - \frac{e^{-3x}}{x}\bigg) \; dx$

Evaluate the integral below $$\int_0^{\infty} \bigg(\frac{e^{-x}}{\sinh(x)} - \frac{e^{-3x}}{x}\bigg) \; dx$$ Using Wolfram I get the integral is $\gamma + \log\bigg(\frac{3}{2}\bigg)$, where ...
2
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0answers
11 views

Fluid Flow: lubrication, integration, ODE

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
1
vote
1answer
23 views

Fourier transform (properties)

I have a function $f$ such that $|f(x)|\leq e^{-x^2/2}$ hence in $\mathcal{L}^2(\mathbb{R})\cap\mathcal{L}^1(\mathbb{R})$ and thus we can compute the Fourier transform $$\hat{f} (\xi) = ...
3
votes
1answer
14 views

Equal integrals, circles, opposite directions

I've found this equality in my complex analysis book, but I don't see why it is true. Could you help me understand it? $$\int _{\partial D(1,1)} \frac{dz}{(z-1)(z+1)} = \int _{- \partial D(-1,1)} ...
1
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1answer
9 views

Finite parameter integral implies finite norm

Need a bit of help with a parameter integral problem. We have, $X$ is a finite measure space with measure $\mu$ and $f:X\rightarrow [0 , \infty)$ is a measurable function. The parameter integral ...
-1
votes
1answer
39 views

Given $f(x)=x+\int_{0}^1 t(t+x)f(t) dt $ , what is $f(0) $? [on hold]

Let $f:\mathbb R \to \mathbb R$ be such that $f(x)=x+\int_{0}^1 t(t+x)f(t) dt $ , then how do we find $f(0) $ ?
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0answers
14 views

Evaluate the line integral with Euler.

Need some help evaluating this line Intergral. $\int$$_c$ xy${e^y}$$^z$ dy Where C: x = 4t ; y = 3t$^2$ ; z = 3t$^3$ ; 0$\le$t$\le$1 Any help would be great. Thanks.
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0answers
24 views

Why does this integration proof not work for proving the formula for the surface area of a cone?

I know there is a similar question to this one. However, the method used there is slightly different; I would like to know what is wrong with my method I tried to prove that the surface area of a ...
4
votes
1answer
49 views
5
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0answers
37 views

closed form for $I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx$

solve $$I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx:0<n<2$$ I tried for $n=1$ : $$I(v)=\int_{0}^{\infty}\frac{x}{x^2+u^2}\tanh(vx) dx$$ ...
1
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1answer
36 views

Proving that $F(x)$ is a constant

This was on a test and i know i was supposed to use 2nd ftoc to prove that $F(x)$ was a constant when $x>0$ $$ F(x) = \int_{0}^{x} \frac{1}{t^2 +1} dt + \int_{0}^{\frac{1}{x}} \frac{1}{t^2 +1} ...
0
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0answers
34 views

Are there ever cases where it's easy to get coefficients for the series representation for an integrand, but hard to approximate the integral?

WHY I'M ASKING THIS I'm working on a faster way to approximate integrals of series. So I'd like to know if this could be useful. THE QUESTION If we suppose that we can get a formula for the ...
7
votes
2answers
83 views

Prove that $\int_0^\infty \frac{\ln x}{x^n-1}\,dx = \left(\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}\right)^2$

This question inspired me to ask the following. Prove that $$I_n = \int_0^\infty \frac{\ln x}{x^n-1}\,dx = \left(\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}\right)^2,$$ for $\Re(n)>1$. For some ...
0
votes
2answers
36 views

Finding the volume of revolution using the method of shells

I'm trying to find the volume of the solid generated by revolving the region bounded by $y=x^2$ and $y=6x+7$ about $x$-axis using the shell method. I applied the method and I got $15864/5$ ...
1
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0answers
15 views

integration with pade approximant

given the function $$ \int _{0}^{\infty}\sqrt{x}exp(-x) $$ can we use Pade approximants to integrate this i mean let bhe te rational approxsiamtions of $ \sqrt{x}= \frac{A(x)}{B(x)} $ and $ ...
0
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0answers
7 views

Another mutivariable integral over a simplex

Let $p$ and $q$ be two positive integers and let $\beta \neq 1/2$ be a real number. Then let $B > A > 0$. With the help of Mathematica, ie by doing elementary integrations and by consecutively ...
4
votes
2answers
49 views

Solving Integrals w/Trig

I need to solve the following integral: $$\int \sin^2(x)\cos^2(x) dx$$ This problem belongs to math notes that can be found here. Here are the steps listed to solve the equation. I can solve to a ...
3
votes
1answer
78 views

Closed form of $I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx$

Does the integral below have a closed-form: $$I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx,$$ where $\tan^{-1} (\cdot)$ is inverse tangent function. ...
0
votes
1answer
5 views

How to find the limits of integration to get the area for a loop of a lemniscate?

I know how to integrate the squared radius to get the equation that'll give me the area, like such for a lemniscate with $r^2=8\sin(2\theta)$ : $$1/2\int 8sin(2\theta) = 4 \int \sin(2\theta) = 4 * ...
1
vote
2answers
46 views

What is $\int_0^{\infty} x^2e^{\frac{(x-\mu)^2}{2 a^2}} dx$?

How can we express the integral $\int_0^{\infty} x^2e^{-\frac{(x-\mu)^2}{2 a^2}} dx$ for example by means of the error function? The problem is of course, that the expectation value is shifted and we ...
1
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0answers
24 views

Can I do the following when solving my integration??

I appreciate any feedback for my question. I have an integration as follows $$\int_{-\pi}^{\pi}\frac{1}{2\pi} \prod_i \frac{1}{1+ x_ig(\theta)} d\theta $$ I have that $g(\theta)$ is the defined as ...
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0answers
379 views

Relation I found: $ (\sum_{r=1}^{\infty}\frac{z(r)}{r})\times \int_0^\infty f(x) dx = \lim_{h \rightarrow 0} \sum_{i=0}^{n} f(k_ih)h$

I was fiddling with some maths and came up with an interesting relationship: $$ (\sum_{r=1}^{\infty}\frac{z(r)}{r})\times \int_0^\infty f(x) dx = \lim_{h \rightarrow 0} \sum_{i=0}^{n} f(k_ih)h$$ ...
8
votes
2answers
104 views

Test for convergence $\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$

What is the easiest way to test the convergence of $$\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$$ Is it possible to only use the high school tools for that?
-1
votes
5answers
58 views

Does $\int_0^{\infty}\frac{x\hspace{1mm}dx}{x^3+1}$ converge? [on hold]

Does $\int_0^{\infty}\dfrac{x\hspace{1mm}dx}{x^3+1}$ converge? Can some explain how to approach this problem? All ideas are appreciated
0
votes
1answer
24 views

what is the order of integration for : integral of x dx * integral of y dy

I'm still trying to get my head around he basics of this stuff so please use simple language in your answer ! $$ \int dx \int f(x,y) dy$$ the first integral limits are from 0 to 1 for dx and the ...
-1
votes
2answers
25 views

Integral Differentiation over constants [on hold]

Let $f(x)$ be integrable on $x\in[0,X]$ and $a,b>0$ constants. I would like to get the derivative of $$I(x)=a\int^x_0{(b-X-f(x))dx}$$ with respect to $b$, i.e. $\dfrac{\partial I(x)}{\partial b}$. ...
0
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0answers
33 views

Is it possible to solve this set of equations?

Let's have system of equations: $$ \tag 1 [\nabla \times \mathbf E ] = -\frac{\partial \mathbf B}{\partial t} , $$ $$ \tag 2 [\nabla \times \mathbf B] = \sigma \mathbf E + A(\mu \mathbf K + C \mathbf ...
0
votes
0answers
31 views

How to solve the following integral [on hold]

Do you have any idea how to solve the following integral: $$\int\limits_0^a {{e^{\large \left(- \frac{{by}}{{c - dy}} - ey\right)}}dy}$$ where $a$, $b$, $c$, $d$ and $e$ are constants? Thank you ...
12
votes
5answers
160 views

Evaluation of $\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$

How to evaluate the following integral $$\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$$ It looks like beta function but Wolfram Alpha cannot evaluate it. So, I computed the numerical value of ...
0
votes
0answers
38 views

A question about $f(x)\equiv C$

Let $f(x)$ is Continuous function on $[0,\pi]$,and for $n=1,2,.....,$ the function $f(x)$ has the following property:$$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0.(n=1,2,......)$$ Proof: $f(x)\equiv C$(C is ...
0
votes
0answers
8 views

Integral over homogeneous function does not vanish

Let $\alpha>0$ be a multi-index. For $x,y\in\mathbb{R}^n$, $n>1$, we consider the integral $$\int_{|x|=1} \int_{|y|=1} \partial_y^\alpha f(y)\ g(x,y)\ \mathrm{d}y \mathrm{d}x\qquad (*)$$ where ...
2
votes
2answers
95 views

How to show that $\int_0^1 dx \frac{1+x^a}{(1+x)^{a+2}} = \frac{1}{a+1}$?

From numerical evidence it appears that whenever the integral converges, $$J_a :=\int_0^1 dx \frac{1+x^a}{(1+x)^{a+2}} = \frac{1}{a+1}.$$ For $a \in \mathbb{N}$, I was able to prove this using ...
1
vote
0answers
20 views

Accelerometer data integration (MMSE)

Based on the raw accelerometer measurements use simple integration on the raw $X$ and $Y$ axis data to determine the velocity and position. If we assume a linear model $Y = aX + b$ for determine the ...
1
vote
2answers
36 views

Integral of pdf

I need to find the integral for this pdf but I don't know if I need to, or can, take the integral of two variables at the same time. $$ f(x;\theta)=\frac{x}{\theta^2} e^{-x^2/(2\theta^2)} ,\quad ...
2
votes
1answer
47 views

If functions converge a.e. and their integrals converge, does convergence in $L^1$ follow?

I was wondering if $f_n, f:\mathbb{R}\rightarrow\mathbb{R}$ are s.t. $f_n\rightarrow f$ pointwise a.e. and $\int f_n\rightarrow \int f$ where integrals are Lebesgue Integrals, is there any Theorem or ...
-2
votes
0answers
17 views

Solve definite integral using complex-variable technique. (Engineering mathematics class) [on hold]

Solve the definite integral using the complex-variable technique.
0
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3answers
52 views

how to integrate $\int_{0}^1 \sqrt{(e^x+e^{-x}+2)} dx $? [on hold]

what to find $\int_{0}^1 \sqrt{(e^x+e^{-x}+2)} dx $ ? Could you give me a hint? Thanks!
-4
votes
1answer
34 views

integrate by parts: $\int \cosh^2(x)dx$ please show solution step by step [on hold]

Integrate by parts: $$\int \cosh^2(x)dx$$ Please show the solution step by step. I actually somehow found my self in a loop solving the integral: = cosh(x) sinh(x) - int (sinh(x) (-sinh(x)) (x) = ...
1
vote
2answers
39 views

Integration with square root in denominator

I am honestly embarrassed to ask this because i feel like i should know how to do this but: $ \int \frac{x}{\sqrt{2x-1}}dx $ Try to use u-substitution please
0
votes
2answers
28 views

Gamma function in $C^{2}$

How can I show that for $x>0$, the Gamma function is at least $C^{2}$? The Gamma function is defined as $$\displaystyle \int^\infty_0 e^{-t}t^{x-1}\ dt$$ For which $x$ is the integrand integrable?
1
vote
2answers
84 views

How to integrate $\int \frac{\sqrt{x}}{x+1}dx$?

How to integrate $$\int \frac{\sqrt{x}}{x+1}dx$$ Can I substitute $x+1$ with $u$?
0
votes
1answer
19 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 ...
0
votes
1answer
18 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 ...
2
votes
0answers
29 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 ...