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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0answers
24 views
0
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1answer
44 views

Find the volume of the solid in $\Bbb R^3$

I need to find the volume of the solid in $\Bbb R^3$. It is bounded by the following: $y=x^2$, $x=y^2$, $z=x+y+21$ and $z=0$. I known that the volume is expressed as follows: $$\iiint 1 \, dV$$ I ...
1
vote
4answers
45 views

Area between three lines/curves

I know this is a very elementary question but I can't make out the answer from the other posts I found in my search. These are three lines, I need to find the area enclosed by them. how do I go ...
3
votes
1answer
53 views

Solving an integral with trig substitution

I'm looking to solve the following integral using substitution: $$\int \frac{dx}{2-\cos x}$$ Let $z=\tan\frac{x}{2}$ Then $dz=\frac 1 2 \sec^2 \frac x 2\,dx$ $$\sin x=\frac{2z}{z^2+1}$$ $$\cos x ...
0
votes
1answer
9 views

Let $f_1 , f_2: I\mapsto \mathbb{R}$ bounded functions. Show that $L(f_1)+L(f_2)\leq L(f_1+f_2)$ (Riemann integral)

Let $f_1 , f_2: I\mapsto \mathbb{R}$ bounded functions. Show that $L(f_1)+L(f_2)\leq L(f_1+f_2)$ where $L(F)$ is the supremum of the lower sums of the Riemann integral. I tried to by contradicction ...
0
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1answer
20 views

Applying the fundamental theorem of Integration

We know from the fundamental theorem of Integration that for a continuous function $ f:[a,b] \rightarrow \mathbb{R} $ with antiderivative $ f:[a,b] \rightarrow \mathbb{R} $ we have that $ ...
3
votes
2answers
62 views

how can I show this integral diverges?

I want to show $E(T_a)=\infty$ $$E(T_a)=\int_0^{\infty}{{x|a|}\over\sqrt{2\pi}}x^{-3/2}e^{-a^2/x}dx$$ to show this I need to show this integral diverges. I know gamma function that $$\Gamma ...
1
vote
2answers
27 views

Iterated integrals in general ( and double integral )

$f:[0,1]\times [0,1]\to\mathbb R,$ defined by $$f(x,y)= \begin{cases}1,\quad \ \ y\in\mathbb R\text{\\}\mathbb Q\\2x,\quad\text{otherwise}\end{cases}$$. $1.1$: $\int_0^1f(x,y)dx$ exists for every ...
4
votes
2answers
46 views

Writing integral in terms of distributions

EDIT (now asking how to write $F$ as distributions, instead of writing the integral in terms of distributions): Let $F$ be the distribution defined by its action on a test function $\phi$ as ...
6
votes
6answers
94 views

Two apparently different antiderivatives of $\frac{1}{2 x}$

What is right way to calculate this integral and why? $$ \int\frac{1}{2x}\text dx $$ I thought, that this substitution is right: $$ t = 2x $$ $$ \text dt = 2\text dx $$ $$ \frac{\text dt}{2} = ...
1
vote
1answer
25 views

Integral equation involving Planck radiation formula

I am stuck in solving the following integral equation: $$\sigma T^4=\pi\int_{\lambda_0}^{\lambda_1}d\lambda W_{\lambda,T}$$ where: ...
0
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1answer
26 views

what is the Convolution the expression [on hold]

what is the convolution of the expression, $\ x(t)*y(-t)\ $ I want to apply it to be written in integral form.
0
votes
1answer
43 views

Fourier coefficients of a symmetric function in $\pi$

I want to show that the Fourier coefficients $\int_{-\pi}^\pi e^{ij \lambda} f'(\lambda) d \lambda$ of the derivative of a continuously differentiable function $f: [ - \pi, \pi] \rightarrow ...
0
votes
0answers
29 views

convergence and holomorphic function - explanations and proofs

I haven't had analysis for a long time and I've forgotten plenty. Could you please explain me and prove that the following converges: $$ \sum_{n \geq 1}e^{-n^2t\pi}, t>0 c $$ and explain why the ...
-1
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0answers
45 views

Could you help me an integral question? [on hold]

Prove that $$ e \int_0^{1/e} (1-x)^{-1/x}\,dx>3.039 $$ where $e=2.71828\dots$ Thanks a lot.
0
votes
2answers
62 views

How to integrate $x/\sqrt{1-x^2}$?

Could somebody please tell me where I made a mistake? I want to integrate \begin{equation*} \int_a^b\frac{x}{\sqrt{1-x^2}}dx. \end{equation*} As far as I know the subsitution $u=1-x^2$ works, but ...
0
votes
1answer
22 views

vector field using green's theorem+other integration

So am I supposed to be using green's theorem for the first question, but where I'm confused is that there are three variables if I do, dx dy dz (I haven't learn how to use green's theorem for 3 ...
1
vote
3answers
98 views

Trigonometric Substitution in $\int _0^{\pi/2}{\frac{ x\cos x}{ 1+\sin^2 x} dx }$

Evaluate $$ \int _{ 0 }^{ \pi /2 }{ \frac { x\cos { (x) } }{ 1+\sin ^{ 2 }{ x } } \ \mathrm{d}x } $$ $$$$ The solution was suggested like this:$$$$ SOLUTION: First of all its, quite obvious to have ...
4
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0answers
37 views

$\int_{-\pi/2}^{\pi/2} \cos(a \cos\theta) e^{im\theta} e^{-ib\sin\theta} \mathrm{d}\theta $ Integration

I am struggling to find the integration of the expression below, $$\int_{-\pi/2}^{\pi/2} \cos(a \cos\theta) e^{im\theta} e^{-ib\sin\theta} \mathrm{d}\theta $$ where $a$ and $b$ are arbitrary constant ...
0
votes
4answers
59 views

Compute Integral of $\int_0^1 e^{\sqrt{x}}dx$ using integration by substitution

Compute Integral of $$\int_0^1 e^{\sqrt{x}}dx$$ using integration by substitution. I will start from: $\int_0^1 e^{\sqrt{x}}dx$ let $u= \sqrt{x}$, $du = -dx\rightarrow dx=-du$ Now I will start ...
1
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0answers
29 views

Integration of gaussian divided by square root of -log(1-x) - does the Meijer G function help me?

After some modelling of my data I came to the following integral: $$ \int_0^{1}\dfrac{exp{\left(-\dfrac{\left(x-\mu\right)^2}{2\,\sigma^2}\right)}}{\sqrt{-\log{(1-x)}}} $$ I cannot solve it, and ...
0
votes
2answers
25 views

evaluating the double integral

I tried to calculate $\int _0^9 dx\:\int _{-\sqrt{x}}^{\sqrt{x}}\:y^2dy$ which yielded $c$ as in this integral has no particular value...when I plot the graphs for it's D however, a certain area does ...
0
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2answers
57 views

Understanding limits that occur in integrals during Feynmann integration

When we use Feynman Integration, how do we decide the final constant of integration? For example, in this problem: $$ f(a) = \int_{0}^{1} \frac{\arctan(ax)}{\sqrt{1-x^2}} dx$$ After ...
0
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0answers
42 views

Feynman Integration

Could somebody please recommend some places to learn Differentiating under the Integral sign? I need to learn this technique as a lot of Integrals can be solved using this. Many thanks!
0
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1answer
34 views

How to calculate the shielding time and determine the time step

The problem is illustrated as follows. A shielding plate scans over a target plate at a constant speed $v_{scan}$ and dynamically shadows the target plate to adjust the exposure time of the light ...
4
votes
0answers
46 views

Difficult definite integral involving Bessel function in the denominator

I came across an integral involving Bessel function in the denominator, derivatives of Bessel functions and complex argument.The equation yields: $$ \int_0^1 ...
0
votes
0answers
29 views

Calculation of a volume integral

Calculate the volume integral $$\int_V z^2r^{-3}e^{-r^2} \, {\rm d}V$$ where $r=\sqrt{x^2+y^2+z^2}$ and $V$ is the whole of $\mathbb{R}^3$. So I want to integrate $$\int_V z^2r^{-3}e^{-r^2} \, {\rm ...
1
vote
1answer
61 views

$\int_{0}^{\frac{\pi}{4}} e^{\sec x} \frac{\sin( x + \frac{\pi}{4})}{(1 - \sin x) \cos x}\, dx$?

How do I find the value of $$ \int_{0}^{\frac{\pi}{4}} e^{\sec x} \dfrac{\sin\Big( x + \dfrac{\pi}{4}\Big)}{(1 - \sin x) \cos x} \;\mathrm{d}x $$
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votes
0answers
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Calc 2 Work Problem [closed]

In a steam engine the pressure P and the volume V of steam satisfy the equation PV^1.4=k, where k is a constant. Use the information given to calculate work done (in ft-lb) by the engine during a ...
2
votes
1answer
55 views

Help with an Inverse Trigonometry Integral 2

Evaluate $$\int^{1/{\sqrt{3}}}_{-1/{\sqrt{3}}} \frac{x^4}{1-x^4}\cos^{-1}\frac{2x}{1+x^2} \mathrm{d}x\\= \frac{\pi}{a}\ln(b+\sqrt{c}) +\frac{\pi^{d}}{e} - \frac{\pi}{\sqrt{f}}$$ Then Find ...
-1
votes
0answers
36 views

Solve $\int_{0}^{\frac{\pi}{2}}\ln^{2}(\sin(x)) \ \mathrm dx$ [duplicate]

I recently had this integration problem on a test and could not solve it any way I tried. How do I do this? $$ \large\displaystyle\int_{0}^{\frac{\pi}{2}}\ln^{2}(\sin(x))\mathrm dx$$
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0answers
51 views

Calculus 2 Work Problem [closed]

In a steam engine the pressure P and the volume V of steam satisfy the equation $PV^{1.4}=k$, where $k$ is a constant. Use the information given to calculate work done in ft-lb by the engine during a ...
2
votes
1answer
76 views

Integral with Logarithms

$$\displaystyle \int _{ 0 }^{ \pi /2 }{ \log(\cos(x))\log(\sin(x)) \ dx } = \dfrac { \pi { \ln}^{ A }(B) }{ C } -\dfrac { { \pi }^{ D } }{ E } $$ $$$$ This was one solution, but it went completely ...
4
votes
1answer
38 views

Simple Derivation of Functional Equation Question (L'Hospital's Rule)

First, the question is: $f$ is a differentiable function and $f : R \rightarrow R$ $xf(x)-yf(y)=(x-y)f(x+y)$ $f'(2x)=?$ My approach for problem is using L'Hospital's rule: $$ ...
1
vote
1answer
39 views

$\int \dfrac{1}{4x^2+1} \ dx $ - why is it $\dfrac{1}{2} \arctan(2x)+c$ not $\arctan(2x)+c$?

$\int \dfrac{1}{4x^2+1} \ dx $ - why is it $\dfrac{1}{2} \arctan(2x)+c$ not $\arctan(2x)+c$? I've been looking at my formula booklet which gives the integral as: $$\int ...
3
votes
2answers
53 views

Evaluating the complex integral $\int_{-\infty}^\infty \frac{\cos(x)}{x+i}\,dx$

I stumbled upon this particular integral a few minutes ago, and I have no idea how to go about it : $$\int_{-\infty}^\infty \frac{\cos(x)}{x+i}\,dx$$ I looked up on the internet and I managed to ...
0
votes
0answers
50 views

What is the extent of the streaks covering a square? [closed]

Let $N$ be a unit square, $1 < A <\sqrt{2}$ real number, we put a strip with width $A$ to the square randomly. I would like to determine the measure of the strips, that cover the square. I don't ...
0
votes
1answer
8 views

I am having difficulty finding, making a triple integral of the space z< |x-y|. Can someone recommend a technique most effective fore such things?

I am having difficulty finding, making a triple integral of the space $z< |x-y|$. Can someone recommend a technique most effective fore such things? I try to somehow draw the graph and see the ...
1
vote
0answers
37 views

Integral of surface

$$\iint_\limits{S}\sqrt{\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}}dS$$ where $$ S: \ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ I tried to solve it : $$\iint_{S^+}P(x,y,z)\ dydz = ...
0
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0answers
35 views

Calculus integration latest

Use a double integral to find the volume of the solid bounded by two surfaces $$x^2+y^2=4,$$ and $$x^2+z^2=4.$$ is it using one way to integrate or there is using separation way?
2
votes
1answer
41 views

Trouble solving extremely simple integration by parts

This is a very basic question so it's kind of embarrassing but I can't seem for the life of me to get the right answer for some reason. I want to find $\int\frac{x}{(1+x)^2}dx=\int ...
0
votes
1answer
41 views

Computing $\int_{0}^1 \frac{\left( (1-x )a +x b\right)^2}{(1-x)c +x d} dx$

I want to find the integral of \begin{align*} \int_{0}^1 \frac{\left( (1-x )a +x b\right)^2}{(1-x)c +x d}dx \end{align*} for any $a,b$ and $c>0$ and $d>0$. Using Wolfram-Alpha I found that ...
1
vote
2answers
115 views

Cannot understand an Integral

$$\displaystyle \int _{ \pi /6 }^{ \pi /3 }{ \frac { dx }{ \sec x+\csc x } } =\frac { \sqrt { a } -b }{ 2 } +\frac { \sqrt { c } }{ 2 } \log(\sqrt { d } +\sqrt { e } -\sqrt { f } -g)$$ I had to solve ...
0
votes
0answers
41 views

Closed form for integral of an error function

My question is similar to that posted here. I have the following integral that I want to determine in a closed form. My uncertainty arises due to the addition term within the Error function: ...
2
votes
2answers
43 views

Asymptotic behaviour of a double sum

I need to find the asymptotic behaviour of the following double sum: $$ S_{n,\alpha,p}:=\sum_{k_1=1}^n\sum_{k_2=1}^n \frac{(k_1k_2)^{p-2}}{(k_1+k_2)^{\alpha p}}, $$ depending on the parameters ...
0
votes
1answer
28 views

Absolute value integral

I think this is trivial but my mind does not work as it should. I have the following sequence of functions: \begin{equation} f_n(x)=\begin{cases} n(1-n|x+2|), & |x+2|<1/n \\ 0, & |x+2|\geq ...
2
votes
2answers
66 views

Find the value of undefinite integral

Find $$\int \frac{dx}{(x+1)^{1/2}+(x+1)^{1/3}}$$ I have tried with let $u=(x+1)^{1/2}+(x+1)^{1/3}$ but I have nothing to solve that undefinite integral. please give me a clue for solve it.
0
votes
1answer
25 views

polar coordinates question

I was tasked with writing $\iint_D f(x,y) \,dx \,dy$ for $ [ D:{4\leq x^2 + y^2 \leq}9]$ through ''reoccurring integrals'' in polar and Cartesian systems? what are ''reoccurring integrals''? and how ...
1
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0answers
28 views

polar system in a plane?

what is a polar system in a plane and how it helps in calculating integrals in certain areas? I'm looking for a good explanation/a fair/ readable source on the matter.
1
vote
1answer
18 views

integral over domain $1_{(x+y≥100)}$

The problem is: Compute: $\frac {1}{40^2}\int_{40}^{80}\int_{40}^{80} (100-x)1_{(x+y≥100)}dxdy$ my attempt: $$=\frac {1}{40^2}\int_{40}^{80}\int_{100-x}^{80} (100-x) dydx = \frac ...