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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1
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3answers
43 views

Why the anti derivative of $\sec(x) \cdot \tan(x)$ is $\sec(x)$?

I have discovered that $$\sec(x) = \frac{1}{\cos(x)}$$ but I do not understand why the indefinite integral of $\sec(x) \cdot \tan(x)$ is $\sec(x)$. I am watching the following videos: ...
1
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0answers
12 views

A real integral (may be requires contour integration)?

The integral I have in mind is $$\int^\infty_0 x^{r}(x + \lambda)^{-1}dx$$ where $r \in (-1, 0)$, and $\lambda$ is a non-negative constant. I apologize if this is really easy and I am missing some ...
1
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0answers
46 views

Geometric proof for Sophomore's dream

Is there a "visual proof" for sophomore's dream? $$\int_0^1 x^{-x}\,dx = \sum_{n=1}^\infty n^{-n}.$$ In the wikipedia article there are two algebraic proofs, but the integral and the summation has ...
0
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1answer
13 views

use laplace transform to solve the given integral equation

use Laplace transform to solve the given integral equation I don't know how start because it differences on other Laplace question I see before
1
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1answer
25 views

curves and integral

Find the area between these curves. $$y=\dfrac{3}{2x+1},\qquad y=3x-2;\qquad x=2\quad \text{et} \quad y=0 $$ indeed, I calculate the integral of the blue function between $1$ and $2$. Then, I ...
0
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0answers
18 views

Integral involving Whittaker function

Consider the following integral: $$ \int_1^{\infty} \frac{e^{u/2}}{u}[-\mathrm{Ei}(-u)]\,W_{1,\imath p}(u)\,du, $$ where $\imath=\sqrt{-1}$ and $p>0$ selected so that $W_{1,\imath p}(1)=0$; here ...
0
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0answers
15 views

Mixture of binomial distributions

I have a population of agents with a single real-valued attribute $x$. Each of them performs $n$ Bernoulli trials with success probability $q(x)$ which depends on their attribute. In particular, $$ ...
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2answers
34 views

A question related to measura space

Let a real value $X$ be a random variable and consider $\int_{\Omega}|X|dP \lt \infty $. I need to show that \begin{equation*} nP(|X|\gt n)\to_{n\to \infty} 0. \end{equation*} please help me ...
2
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1answer
66 views

Study the following integral: $\int_0^\infty \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x … (\ln^{(k)} x)^s }$

How do I calculate for which values of $s$ the following integral converges? $$\int\limits_{0}^{\infty} \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x \cdots (\ln^{(k)} ...
1
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0answers
36 views

Triple integration, sphere (electric field).

The sphere $K\subseteq R^3$ with radius $R$ has a homogeneous charge density $\rho$. Find the electric field E, produced by K outside of, meaning, find the integral ...
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0answers
17 views

The area of a stereographic projection

I'm newbie at multidimensional integration and I'm trying to make a working algorithm in Wolfram that can help me compute the areas on the unit sphere without complicated parametrization, provided I ...
1
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0answers
32 views

Calculating total mass of a wire

I'm giving the following $$ \delta(x) = x + 7,\quad (0 \leq x \leq 4) $$ It says you are given the length-density function, $\delta(x)$, of an ininfinitesimally thin wire lying on the $x$-axis over ...
2
votes
4answers
338 views

How to solve certain types of integrals

I'm asking for a walk through of integrals in the form: $$\int \frac{a(x)}{b(x)}\,dx$$ where both $a(x)$ and $b(x)$ are polynomials in their lowest terms. For instance $$\int ...
2
votes
1answer
34 views

Evaluating a triple integral by inspection

I would like to evaluate the triple integral: $$\iiint\limits_D {2 + 3{x^2} + 3{y^2}dV}$$ where $D$ is a conic domain with vertex $(0,0,b)$ and axis along the $z$-axis with a base (disk) with radius ...
0
votes
1answer
24 views

Finding the complex Fourier series… [on hold]

Here's is the problem: And this is my solution for part (i) of the problem. Is it correct? Can I simplify it more than I did? For part (ii), I don't know how to start. So please give me a ...
-4
votes
1answer
47 views

How can I calculate this integral? [on hold]

Calculate the integral: \begin{equation*} \lim_{n\rightarrow\infty}\int_{0}^{2\pi} \frac{\ cos\ nt} {\ cos\ (t)+4} dt. \end{equation*}
5
votes
2answers
114 views

Infinite integrals$\int_0^{ + \infty } {\frac{1}{{\left( {x + 1} \right)\left( {{x^n} + 1} \right)}}dx} .$

How to calculate $$\int_0^{ + \infty } {\frac{1}{{\left( {x + 1} \right)\left( {{x^n} + 1} \right)}}dx} .$$
2
votes
2answers
22 views

Cylinder cut out of a sphere. (volume).

1.) A cylinder $Z=${$(x,y,z)\in R^3|x^2+y^2\leq \rho^2$}$~~$ ($0<\rho<R$) is cut out of a sphere $K\in R^3$ with radius $R>0$ centered around the origin. Find the volume of the rest ...
-2
votes
1answer
39 views

Integration by parts prove integral of cos^n x dx [on hold]

I'm having a problem with one of my questions. How can I prove that $\begin{align}\int\cos^n x dx&=\sin x\cdot\cos^{n-1}x+(n-1)\int\sin^2x\cos^{n-2}x dx\end{align}$ ?
4
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1answer
25 views

Double integral (and for the area enclosed by a lemniscate).

a) Transform into polar coordinates and compute the integral $\int_\Omega ln(1+x^2+y^2)d(x,y)$ where $\Omega$ is the interior of the unit circle in the first quadrant. b) $\Omega\subseteq R^2$ ...
2
votes
0answers
44 views

What can the various ways of integrating $\int \frac { x ^2 }{ (x \sin (x) +\cos(x))^2} \, dx$

$$ \int \frac {x^2}{(x\sin(x) + \cos(x))^2} \, \mathrm{d}x $$ Well I found a method for solving this sum in a book saying that : We can multiply and divide the expression by $x\cos(x)$ and then ...
0
votes
0answers
22 views

Can this definite integral of an inverse Laplace transform by simplified?

Can either of the below expressions involving an unknown analytic function $h(s,t)$ and the inverse Laplace transform $\mathcal{L}^{-1}$ be simplified? $$ \int\limits_{0}^1 \mathcal{L}^{-1} \left\{ ...
0
votes
3answers
52 views

integration $\int_{0}^{1/8} \frac{4}{\sqrt{(1-4x^2)}} \,dx$

integration equation $$\int_{0}^{1/8} \frac{4}{\sqrt{(1-4x^2)}} \,dx$$ my work $t= \sqrt{(1-4x^2)} $ $dt = -4x/\sqrt{(1-4x^2)} dx $ stuck here also
0
votes
2answers
40 views

I need help with the integration order please

the integral is as follows: find the volume between these regions bounded by : $z = x^2 + 3y^2$ and $z = 9 - x^2$ I discovered that this would be the space bounded by the elliptic paraboloid and the ...
-1
votes
2answers
34 views

solve $(x^2 y+y)dy/dx =\arctan(x)$ , $y(1) = -1$

solve the equation $(x^2 y+y)(dy/dx) = \arctan(x)$ , $y(1) = -1$ my work until now $(x^2 y+y)\,dy = \arctan(x)\,dx$ $x^2 y\,dy+y\,dy = \arctan(x)\,dx$ ---> stuck here
1
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3answers
38 views

Integral Question using the Rule of Subsitution

I'm confused as to why $ \int e^{kx}dx$ = $\frac{e^{kx}}{k} + C$. I'm using the rule of substitution and came to the conclusion that it should be $e^{kx}k$ because the derivative of $kx$ is $k$. What ...
0
votes
0answers
27 views

Gradient and Laplacian in integral.

Let $u,v,f$ be functions of $\mathbb{R}^n$ to $\mathbb{R}$, with compact support in a domain $U$, this formula $$\int_{U} f(x) (Du \cdot Dv) dx = \int_{U} f(x)(u D(Dv)) dx = \int_{U} f(x) u(x) \Delta ...
0
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0answers
45 views

Calculate integral

I have to calculate $$\operatorname{PV}\int_{-\infty}^{\infty}\frac{1}{\pi}\frac{y}{1+y^2}dy$$ I ended up with ...
0
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0answers
17 views

Using polar coordinates in this integral

I'm trying to solve something along the lines of: $$\iint \frac{\partial F_1(x,y)}{\partial x}+\frac{\partial F_2(x,y)}{\partial y}dydx$$ which I want to change to polar coordinates, but I don't ...
2
votes
4answers
57 views

integration $\int_{0}^{3}(12/(x^2 -6x+12))\,dx$

$$\int_{0}^{3} \frac{12}{x^2 - 6x + 12} \,dx$$ I assume that $x^2 - 6x + 12 = (x-3)^2 + 3$, then $t = x - 3 \rightarrow dt = dx$ since $$\int_{0}^{3} \frac{12}{t^2 + 3}\,dt$$ and now I am stuck. ...
1
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1answer
75 views

Feynman Integration Problem

$$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx $$ Evaluate $I$ $$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx$$ $$f(a) = \int_0^1 ...
3
votes
2answers
61 views

$\int \frac{\exp (z)(\sin(3z)}{(z^2-2)(z^2)} dz$ on $|z|=1$

So I need to calculate \begin{equation*} \int \frac{\exp(z) \sin(3z)}{(z^2-2)z^2} \, dz~\text{on}~|z|=1. \end{equation*} So I have found the singularities and residues and observed that the ...
0
votes
0answers
24 views

Can limits to definite integral be vectors?

There are two cases: 1-Limits are scalar and function to be integrated is vector. 2- Limits are vector and function to be integrated is vector. Are both valid. If yes can you give example for each. ...
2
votes
1answer
22 views

Calculating volume by shell integration

$y = \ln x$, region is delimited by $y = -1$, $y = 2$ and the $y$-axis, it rotates around the $y$-axis. It's quite simple to solve by using disk integration but I can't get it right with shell ...
0
votes
2answers
63 views

Why does $\int_0^{\infty}e^{-\frac{z^2}{4}}dz=\sqrt{\pi}\ $?

Why does $$\int_0^{\infty}e^{-\frac{z^2}{4}}dz=\sqrt{\pi}$$ How do you get there from the standard integral which equals $\sqrt{2\pi}$ ?
1
vote
2answers
48 views

Given $|f(x)|=1$,how to construct an $f(x)$, such that $\int ^{+\infty }_{0}f\left( x\right) dx$ converges

Here's the problem: Given $|f(x)| = 1$, construct an $f(x)$, such that $$\int ^{+\infty }_{0}f\left( x\right) dx$$ converges. I think this problem may be done by dividing the 1s and -1s smartly, but ...
0
votes
1answer
18 views

Trying to integrate the volume of a body

I was trying to integrate the volume of a body blocked by $z=0$, $z=2x$, $x+y=3$ and $y=0$ using the double integral... but I don't really know how to approach this.
1
vote
3answers
39 views

A really basic integration question concerning differentials

I'm really, really confused with this. Please, please help me. $$$$ My Calculus teacher taught me that the integral symbol and the differential with respect to which we are integrating are like ...
1
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0answers
34 views

Integration of a symmetric function

I have a bit of confusion about the following situation. Let's assume that we have a symmetric function $f(x,y)$ where it has the property $f(x,y) = f(y,x)$ for all $x$ and $y$. $x$ and $y$ have the ...
0
votes
2answers
38 views

Volume by double or triple integral?

I was trying to integrate the volume of a body blocked by $z=0,\; z=2x,\; x+y = 3$ and $y=0$ using the double integral... however it didn't work yet. I'm convinced its a double integral and not a ...
0
votes
2answers
37 views

Can you explain this equation to someone who hasn't learned integration?

So I have just entered 11th grade and started limits on my own but my Physics textbook has an equation which I don't understand, I suspect it uses integration which I haven't learned yet. So can ...
2
votes
1answer
71 views

A necessary condition to $F'(x)=f(x)$ for a continuous function $f$

Theorem: Consider , $$F(x)=\int_a^xf(t)\,dt$$ If the function $f:[a,b]\to \mathbb R$ is continuous then $F(x)$ is differentiable and $F'(x)=f(x).$ I know that the continuity condition ...
3
votes
2answers
71 views

Calculating $\int_0^\pi \frac{1}{a+b\sin^2(x)} dx $

How do I calculate this integral? $a \gt b$ is given. $$\int_0^\pi \frac{1}{a+b\sin^2(x)} dx $$ I am confused since WolframAlpha says one the one hand, that $F(\pi) = F(0) = 0 $ , but with some ...
-3
votes
2answers
44 views

Differential equations/ 4 [on hold]

How to solve this differential equation: $$ \frac{ydy+zdz}{\sqrt{y^2+z^2}}+\frac{ydz-zdy}{y^2}=0$$ I gave similar but then nothing happens maybe this is exact differential equations?
0
votes
1answer
27 views

Partial derivatives after a change of variables

Say I have a function of $n$ variables $F(x_{1}, x_{2}, x_{3},...,x_{n})$, where $x_{1} = g_{1}(y_{1}, y_{2}, y_{3},...,y_{m})$, $x_{2} = g_{2}(y_{1}, y_{2}, y_{3},...,y_{m}),\dots, x_{n} = ...
2
votes
1answer
46 views

Verify solution to ODE

I am given the ODE $$\left(f''(x)+\frac{f'(x)}{x} \right) \left(1+f'(x)^2 \right) = f'(x)^2f''(x)$$ and I already know that the solution to this ODE is given by $$f(x)= c \cdot arcosh \left( ...
-1
votes
1answer
64 views

How do I calculate $ \int_{-\infty}^{\infty} e^{-ax^2} \;\mathrm{d}x$ for $a>0 $ [duplicate]

$$ \int_{-\infty}^{\infty} e^{-ax^2} \;\mathrm{d}x \quad\text{for } a>0 $$ I don't even know where to begin to be honest so I haven't made any progress on it. The answer is $\sqrt{\pi/a}$. Is ...
0
votes
0answers
19 views

solution uniqueness of non-linear Fredholm equations

the equation is $F(x)=G(\int k(x,y)f(y)dy)$ $(*)$ where $f(x)=dF(x)/dx$ is the unknown and it's required to be non-negative. With integral by parts we'll have the form of a non-linear Fredholm ...
1
vote
1answer
24 views

Cartesian to Spherical Coordinates

Can someone help me passing this triple integral from cartesian to spherical coordinates ? Thanks in advance ;) ...
1
vote
1answer
49 views

Questions about integration

I'm still a bit confused about definite integration although got the basic idea of how to do integration. The problem is to integrate functions on a uniform distribution over [50, 150]. Firstly ...