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
5 views

Stoke's Theorem Application on Cylinder

This is a question regarding Stoke's theorem's application. This is in regards to a problem from MIT OCW. My question is, referring to the answer provided, what closed surfaces are used in the proof ...
1
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0answers
14 views

How to take derivative of $F(u)=\sum_{i=1}^{N} \int f^2(x) u_i^q(x) dx $

I have to find the derivative of a function. Could you help me to find it $$F(u,x)=\sum_{i=1}^{N} \int_{\Omega} f^2(x) u_i^q(x) dx $$ where $q \ge 1$, $f(x): \Omega \to R$, $u_i$ is membership ...
1
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0answers
20 views

How to evaluate the integral ${\displaystyle\int_0^{m_1} \int_0^{m_2} }\frac{dx' dy'}{[(x-x')^2+(y-y')^2+25]^{\frac{3}{2}}} $

How to evaluate the integral $$\int_0^{m_1} \int_0^{m_2} \frac{dx' dy'}{[(x-x')^2+(y-y')^2+25]^{\frac{3}{2}}} $$
0
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0answers
16 views

Interchange Order of Integrals

Can someone explain the last step in this process. Specifically, how do you get the new limits of integration? Expected Value Definition: $E[Y] = \int_0^\infty{P\{Y \ge y\} \, dy}$ Expand: $E[Y] = ...
2
votes
3answers
68 views

Integrate $\displaystyle \int \sin(\sqrt{at})dt$

Integrate $\displaystyle \int \sin(\sqrt{at})dt$ Here is what I tried. Let $u=\sqrt{at}$, then $\displaystyle\ du=\frac{a}{2\sqrt{at}}dt=\frac{a}{2u}dt\implies \frac{2udu}{a}=dt.$ So by subsitution, ...
0
votes
1answer
16 views

Derive $E(X^k)$ I need help with the substitution piece.

If $X\sim\mathrm{WEI}(\theta,\beta)$, derive $E(X^k)$ assuming $k > -\beta$. Note that $X\sim\mathrm{WEI}(\theta,\beta)=\dfrac{\beta}{\theta^\beta}x^{\beta -1}e^{-(x/\theta)^\beta}$ I know to ...
8
votes
2answers
113 views

How to find $I=\int_{-4}^4\int_{-3}^3 \int_{-2}^2 \int_{-1}^1 \frac{x_1-x_2+x_3{-}x_4}{x_1+x_2+x_3+x_4} \, dx_1 \, dx_2 \, dx_3 \, dx_4$

How can I find this integral $$I=\int_{-4}^4\int_{-3}^3 \int_{-2}^2 \int_{-1}^1 \frac{x_1-x_2+x_3{-}x_4}{x_1+x_2+x_3+x_4} \, dx_1 \, dx_2 \, dx_3 \, dx_4$$
0
votes
3answers
68 views

Evaluate the following integration below [on hold]

Evaluate the following integration $$\int_{0}^{\infty }\frac{x^2}{x^6 +1}dx$$ help guys please, I tried but I got nothing.
0
votes
1answer
98 views

Integrating $\int{\frac{x^2}{1+x^5}dx}$ [on hold]

I just encountered the following integral $$\int{\frac{x^2}{1+x^5}dx}$$ At first it appeared to be simple, but I don't know how to solve it. Please share any ideas.
2
votes
2answers
43 views

Definite Integration, keep getting wrong answer.

Correct to 4 significant figures $$\int_{1}^{2}{\csc^24tdt}$$ Done this multiple times now and can't seem to get the answer at the back of the book. Here's my attempt: ...
1
vote
0answers
28 views

Integral of $|\cos(ax))|\times e^{-x^2/b}$

I can compute the following integral very easily ($a$ and $b$ are real and positive): $$\int_{-\infty}^{\infty} \cos(ax)\times \frac{1}{\sqrt{\pi b}}\cdot e^{-\frac{x^2}{b}}\,dx = ...
-1
votes
3answers
20 views

Area bound by two curves. [on hold]

What is the area of the region bounded by the curves $y= 2x^2+7x$ and $y= 2/x$ between $x= 1$ and $x= 3$? Thank you!
6
votes
2answers
40 views

closed form for $\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{dx}{(b^2+x^2)}$

closed form for : $$\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{\mathrm{dx}}{(b^2+x^2)}$$ where $\beta$ is beta function I tried with the definition of beta and i got ...
0
votes
0answers
18 views

Showing certain sum as a Riemann-Stieltjes integral

Let $e(\beta) = e^{2 \pi i \beta}$. I am reading an article, where the author defines the following sum $$ S(N) = \sum_{0 \leq x \leq N, x \equiv g (mod \ q)} \Lambda(x) e(f(x) \alpha), $$ where $f$ ...
1
vote
3answers
90 views

Simplest way to integrate this expression : $\int_{-\infty}^{+\infty} e^{-x^2/2} dx$ [duplicate]

I'm toying around with statistics and calculus for a project of mine and I'm trying to find the simplest/fastest way to integrate this formula : $$\int_{-\infty}^{+\infty} e^{-x^2/2} dx$$ I do not ...
-1
votes
1answer
44 views

Proving that the integrals of two functions are the same if they are equal everywhere except a point [on hold]

Let $f(x)$ and $g(x)$ be integrable functions over $[a,b]$ and let $∂$ be a point on $[a,b]$. If $f(x) = g(x)$ for all $x≠∂$, then $$\int_a^b f(x)dx=\int_a^b g(x)dx$$
1
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3answers
50 views

How do i find this : $\int \frac{1}{(x+a) \sqrt{x+b}}\ dx$, where $a > b > 0$?

Is there someone show me how do I find : $$\int \frac{1}{(x+a) \sqrt{x+b}}\ dx$$, where $$a > b > 0$$ ? I tried to make it as sum of fraction to be easier but sorry i didn't up Thank you for ...
0
votes
0answers
14 views

Implicit numerical integration: error bound

Suppose I'm solving this equation numerically with a time step $h$: $$x''(t) = f(x)$$ Discretizing it and using implicit integration: $$x^{n+1} - 2x^n + x^{n-1} = h^2f( x^{n+1})$$ $x^{n-1}$ and ...
2
votes
7answers
119 views

Evaluating the indefinite integral $\int e^{-x^2}\,\mathrm{d}x$ [on hold]

In my book, it is said that $$\int e^{-x^2} \, \mathrm{d}x$$ cannot be solved by the method of inspection. It then turned to method of substitution as a new topic. I am not able to solve this ...
6
votes
3answers
75 views

How do I evaluate this : $\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx $ for $a > 0$?

How do I evaluate this integral if I supposed that : $a > 0$ $$\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx .$$ For $a=2$ I have got : $2\pi$ I think the result will be : ...
0
votes
1answer
59 views

Calculus 2 - $\int(\sqrt{72+36x^2}dx$

I have done this problem several times and this is the only answer i ever come to. My schools webwork gives me incorrect for my answer (answer is not simplified but it should be accepted in this ...
-2
votes
1answer
47 views

Riemann integrability of the function that is equal to $0$ only at $1,1/2,1/3,…$ and $1$ otherwise

How can I prove that the following function is Riemann Integrable on [0,1] using the criterion $U(P,f)-L(P,f)< \varepsilon$ \begin{equation} f(x)=\begin{cases} 0 & \text{if } x=1, 1/2, 1/3, ...
1
vote
1answer
24 views

Find the Fourier coefficients of $g(x)$

Let $f:\mathbb{R}\to\mathbb{C}$, $2\pi$ periodic function and $f\in C^1$, such that the n-th Fourier coefficient is: $\hat{f}(n) = 3^{-n^2}$. Find the Fourier coefficients of $g(x) = \pi ...
1
vote
2answers
71 views

How do I evaluate this integral $I = \int_{0}^{2 \pi} \ln (\sin x +\sqrt{1+\sin^2 x}) dx$?

I used some variables change to evaluate this integral but i'm not succeed may I have some wrong step as trigono-transformation.Then Is there some one who can show me how do evaluate this : $$I = ...
0
votes
2answers
55 views

Find the area of the entire region that lies between $r=1+\sin\theta; r=1+\cos\theta$

I have to find the area of the region that lies between the curves $r=1+\sin\theta; r=1+\cos\theta$ . The answer the book gave was $\frac {3\pi}{2}-2\sqrt{2}$ . I tried generating the curve for ...
0
votes
1answer
19 views

Derivative of an integral over a varying domain

Consider the function $$H(\alpha) = \int_{\Omega(\alpha)} h(\alpha,x) dx,$$ where $\alpha\in\mathbb{R}$ and $\Omega(\alpha)\subset\mathbb{R}^2$ is a domain that varies continuously with $\alpha$. Is ...
1
vote
1answer
19 views

Error bound of midpoint rules with unbounded second derivative

It is well known that error bound of midpoint rule for function $f[a,b]$ is given by $$ E\leq K\frac{(b-a)^3}{24 n^2} $$ where $|f(x)''\leq K|$ and $n$ is the number of time steps. if second ...
9
votes
3answers
93 views

What's happening at $a=-1$ in $\int x^a dx$? [duplicate]

If we take the right limit $$\lim_{a\to-1}\int x^a dx=\lim_{a\to-1}\frac{x^{a+1}}{a+1}=+\infty$$ but on the other hand $$\int\lim_{a\to-1} x^a dx=\ln x$$ I'm aware you can't just commute the ...
4
votes
1answer
67 views

Area under tangent to a curve.

The tangent to the graph of the function $y=f(x)$ at the point with abscissa $x=a$ forms with the line $x$-axis an angle $\frac{\pi}{6}$ and at the point with abscissa $x=b$ an angle of ...
1
vote
2answers
59 views

Integral of $x^2\sqrt{5+x}\ dx$

I have the following integral to solve, with my working out below. This is a bit more complicated than I am used to, so I'm hoping for some feedback as I'm not sure if my process & solution are ...
2
votes
3answers
59 views

derivative integral $\int_0^{x^2} \sin(t^2)dt$

I want to know how I derivative this integral: $$\int_0^{x^2} \sin(t^2)dt$$ what are the steps to derivative it?
4
votes
1answer
65 views

Find all functions such that $\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$

Is it possible to find all functions such that $$\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$$? My teacher asked us to give examples to prove that this is not true but I was ...
0
votes
0answers
33 views

Show the integration with a complex variable

I want to show that there exists inverse Laplace transform, $f(t)$ of the function $F(\lambda)$. In other word, given $F(\lambda)$, existence of function $f(t)$ such that $$ ...
2
votes
2answers
60 views

Finding $\int_{0}^{\frac{\pi}{2}}\frac{1}{\cos (x-\frac{\pi}{3}).\cos (x-\frac{\pi}{6})}\mathrm{d}x$

How can I find $$\int_{0}^{\frac{\pi}{2}}\frac{1}{\cos (x-\frac{\pi}{3}).\cos (x-\frac{\pi}{6})}\mathrm{d}x$$ ? I suspect this has something simple to do with the basic definite integral properties; ...
0
votes
3answers
57 views

Use the comparison test to find whether $\int_0^\infty 1/(x^2+1)^2\,dx$ converges or not

I was thinking what function I should compare it to. If I say whether a function is smaller or bigger than this one, then I must prove that. I was thinking of (x+1)^2 but I realized that this ...
2
votes
1answer
58 views

Find the area bounded between $f(x)=\frac{\arctan(x)}{x^2}$ and $g(x)=\frac{\arctan(x)}{x^2+1}$

Find the area bounded between $$f(x)=\frac{\arctan(x)}{x^2} \quad\text{and}\quad g(x)=\frac{\arctan(x)}{x^2+1}.$$ The title says the question. The limits are from 1 to infinity. I know that I ...
1
vote
1answer
43 views

Find the SA of a torus

I have been trying to find the surface area of the torus generated by the rotation of $(x-R)^2 + y^2 = r^2$ about the y axis. I tried to use the equation: $$\int_a^b2\pi y\sqrt{1+\left(\frac ...
11
votes
1answer
118 views

Evaluating $\int{ \frac{x^n}{1 + x + \frac{x^2}{2} + \cdots + \frac{x^n}{n!}}}dx$

I tried, and I don't know why it won't work. I used Pascal inversion, I will soon post what I tried. This is mine resolution, I have no clue why it failed. Let's start: define $$I_n(m) = ...
1
vote
2answers
77 views

Finding $\int_{0}^{e^2}(\frac{1}{\log{x}}-(\frac{1}{\log x})^2).\mathrm{d}x$

Finding $$\int_{0}^{e^2}(\frac{1}{\log{x}}-(\frac{1}{\log x})^2).\mathrm{d}x$$ I came upon this problem online, and the answer is given to be $(\frac{e^2}{2}) - e$. However, Wolfram Alpha states that ...
3
votes
1answer
42 views

Probability of True Positive of a random variable defined by an integral expression

$\newcommand{\Prob}{\operatorname{Prob}}$Let's assume that we have a random variable with the following pdf: \begin{equation} f_T(x) = \int_0^\infty f_T(x,g) \cdot f_{g}(g) \, dg = \int_0^\infty ...
2
votes
3answers
67 views

$I_{2n}=\dfrac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq 1$

let $$I_n=\int_0^{\frac{\pi}{2}}\cos^n(t) \, dt$$ show that $$I_{2n}=\frac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq ...
0
votes
0answers
24 views

Volume bounded by $y = \sqrt{25-x^2}, y=0, x=2, x=4$ about the $x$-axis [on hold]

I have come up with this integral $$2\pi \int_3^{\sqrt{21}} y\left(4-\sqrt{25-y^2} \right)\, \mathrm{d}y + 2\pi \int_0^3 2y\, \mathrm{d}y$$ ...
2
votes
2answers
51 views

derivative of a function including a vector

given a column vector including function of a parameters $x=\bigg(f(\beta_1),\ldots,f(\beta_m)\bigg)^T$ where $T$ denotes transpose of the vector. Can somebody tells me what is the derivative with ...
-2
votes
0answers
47 views

Find a triple integral using spherical coordinates.

Solve the following integral: $$\iiint_{V} \mathrm{d}x\: \mathrm{d}y \:\mathrm{d}z$$ Where $V$ is part of the sphere $x^{2}+y^{2}+z^{2}=5$ which is above the $xy$ plane and inside the $x^2+y^2=1$ ...
1
vote
1answer
16 views

Evaluating the volume of a torus formed by rotating a region about a horizontal axis using shells.

Using the method of cylindrical shells, find the volume of the shape created by revolving the region $x^2+(y-5)^2=4$ about $y=-1$. A cylindrical shell is given by: $2\pi v f(v) \ dv$ I solve ...
12
votes
1answer
184 views

Calculating 2 integrals in polylogarithmic functions

Are we aware of any nice way of calculating these $2$ integrals? $$i) \space \int_0^1 \frac{\text{Li}_2\left(x-x^2\right)}{x^2-x+1} \, dx$$ $$ii)\space \int_0^1 ...
-2
votes
0answers
49 views

Is there a way to solve this integral? $\int_0^4 \frac {120x^ky^{4-k}}{\Gamma(5-k)\Gamma(1+k)}dk$ [on hold]

Is there a way to evaluate this integral? $$\int_0^4 \frac {120x^ky^{4-k}}{\Gamma(5-k)\Gamma(1+k)}dk$$
2
votes
2answers
25 views

$F=w(x)=\frac{k}{x^2}$ How much work required to lift a satellite an “infinite distance” into outer space?

The satellite is 6000 lbs at earth's surface, a distance $R$ from the earth's centre (so the answer will be in terms of $R$). I know that it's supposed to be an improper integral going from 0 (?) to ...
0
votes
0answers
13 views

Proof of sets to integrate over in Green's, Stokes and Divergence Theorems

As the title says I've some doubts about these theorems, because my definitions of them are so abstract, I'm struggling to apply them. For Green's theorem, in order to be able to apply it, the region ...
0
votes
1answer
14 views

Evaluate the volume of a solid of revolution using shells.

A cylindrical shell $S$ formed by some revolution about the $y$-axis is given by the equation: $S=2\pi x f(x)dx$, where the circumference $C$ of the shell is $C=2\pi x$, the height of the shell ($H$) ...