Questions tagged [integration]
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 the integrals being considered. This tag often goes along with the (calculus) tag.
64,772
questions
1
vote
0answers
23 views
Calculate $\int_{0}^{1} \int_{y}^{1} e^{-x^2} \, \mathrm{d} x \mathrm{d}y$
Calculate $\displaystyle \int_{0}^{1} \int_{y}^{1} e^{-x^2} \, \mathrm{d} x \mathrm{d}y $
I understand that multiple integral has to be used, but I can't go further.
Can anyone show me how to solve ...
0
votes
1answer
22 views
Find the Area under $R$ Using Integrals
$R$ is a region in the first octant of $\mathbb{R^3}$ that is bounded by $x^2+y^2+z^2=4$ and $z=\sqrt{x^2+y^2}$. Set up the integral for the volume of this region $R$.
I think the first thing to do is ...
1
vote
0answers
15 views
Solution to $\int\frac{1}{x+xe^{ax+b}}dx$
I need to solve $\int\frac{1}{x+xe^{ax+b}}dx$ as part of a larger optimization problem, where the $a$ and $b$ parameters will be optimized in a subsequent step. Wolfram alpha returns the following ...
1
vote
0answers
6 views
U-substitution in proof of Frullani's Theorem
In a proof of Frullani's Theorem I'm trying to understand there's a step where
$$\int_x^y \frac{f(at)}{t}dt - \int_x^y \frac{f(bt)}{t}dt = \int_{ax}^{ay} \frac{f(u)}{u/a} \frac{du}{a} - \int_{bx}^{by} ...
1
vote
1answer
31 views
Not sure how to finish this integral $\int_{-\infty}^\infty x^3 \delta(x^2-2)dx$
Dirac delta is a symmetric function defined as $$\int_{-\infty}^\infty f(t)\delta(t-A)dt = f(A)$$
Find the value of $$\int_{-\infty}^\infty x^3 \delta(x^2-2)dx$$
SOLUTION:
Let $t=x^2$ then $dt=2xdx \...
2
votes
0answers
30 views
Is it possible to get a closed form expression for $E\left(\dfrac{1}{a+e^Z}\right)$ for $Z\sim N(0,1)$ and $a>0$?
The question is in the title. So all we need to do is compute the integral $$\int_{-\infty}^\infty \dfrac{e^{-x^2/2}}{a+e^x}dx$$ I tried some integration by parts tricks but could not proceed. To be ...
0
votes
0answers
38 views
Proof that $\int f(x-a)\cdot g(x)dx = \int f(x)\cdot g(x+a) dx$
How can one proof that:
\begin{equation}
\int f(x-a)\cdot g(x)dx = \int f(x)\cdot g(x+a) dx
\end{equation}
and on what boundaries would this hold (I assume only $[-\infty, \infty]$)?
0
votes
0answers
22 views
Different integral notations
In MS Word there are 2 different integral notations as shown in the screenshot below:
Is there any difference or it is the same?
0
votes
0answers
33 views
Solving for real values of a quadriatic equation
I have already fit a quadratic curve $ax^2 + bx + c$ to experimental data and calculated the value of its coefficients (a, b, c).
Now, I have to compute two integral functions (#14 and #17) as shown ...
0
votes
0answers
23 views
All the functions that satisfy an integral condition
I need to find a caractherize the function $A(a, b)$ such that
\begin{align}
\int_0^xA(x+y, z-\frac{x+y}{2})z dz+\int_0^yA(x+y, z-\frac{x+y}{2})z dz=\alpha x^2+\alpha y^2+\beta xy,
\end{align}
where $\...
0
votes
0answers
28 views
How to solve this integral depending on parameter
$g(\alpha )=\int_{a}^{b} \frac{f(x)}{\sqrt{\left | x-\alpha \right | } } dx$
Given that f(x) is bounded, the question asked me to prove g(α) is continuous on R. The question is, I don't even know if ...
-1
votes
0answers
22 views
integration of product of Exponential and square of Bessel function and x
I need solution of above integral
0
votes
0answers
24 views
Representation of a mollified Coulomb kernel through integration over a sphere
I am looking for a result that allows represent a function through the integral over a sphere. So, does anybody knows a proof of the following statement?
Given $R>0$ we define for every $x\in \...
1
vote
0answers
35 views
Why can't I use Cauchy-Schwarz to show convergence of the Fourier series in $L^1$?
Recently I read that there are $L^1(\mathbb{T})$ functions such that their Fourier series does not converge in the norm of $L^1$, that is if $S_N(f)(x)$ represents the $N$-nth partial sum of the ...
1
vote
0answers
46 views
Double integral $\frac{1}{\sqrt{x^2+y^2}} e^{i(ax+by)}$
How can I integrate
$$\int \int \frac{1}{\sqrt{x^2+y^2}} e^{i(ax+by)} dx dy$$
over $\mathbb{R}^2$, where $a, b$ are constants?
1
vote
0answers
50 views
Integration of differential form inclusion of $S^2$
Take this example. We have the natural inclusion $i : S^2 \rightarrow R^3$ and the differential form: $\omega = x dy \wedge dx + y dz \wedge dx + z dx \wedge dy $. How can we say that that the ...
0
votes
0answers
32 views
Why is this curve the integral curve of the described vector field? [closed]
In Lee's booka bout manifolds we have this example:
I am trying to udnerstand how we can so easily jump from $x^2\frac{\partial}{\partial x}$ to $\gamma(t)$.
2
votes
1answer
68 views
Calculate $\int_{C} e^{\frac{z^2}{z^2-1}}dz$
Calculate $$\int_{C} e^{\frac{z^2}{z^2-1}}dz$$ where $C$ is the circle of radius $3$ centered at $0$ oriented counter clockwise.
So I want to do this by finding $\operatorname{Res}(f,\infty)=-\...
3
votes
2answers
127 views
How can i evaluate this integral with or without using CAS $\int_0^{\infty}\frac{\mathrm dx}{(x^2+1)\cosh(\pi x)}$
$$\int_0^{\infty}\frac{\mathrm dx}{(x^2+1)\cosh(\pi x)}$$
Firstly i used substitution $\pi x=t; \mathrm dx=\frac{\mathrm dt}{\pi}$
$$\pi \int_0^{\infty}\frac{\mathrm dt}{(t^2+\pi^2)\cosh t}$$
writing $...
1
vote
1answer
29 views
Prove that a bounded $f$ is integrable if $I_0 := \lim_{n\to\infty}L(f,P_n) = \lim_{n\to\infty}U(f,P_n)$
Prove that a bounded function $f$ is integrable on $[0,1]$ if
$$I_0 := \lim_{n\to\infty}L(f,P_n) = \lim_{n\to\infty}U(f,P_n),$$ in which case $\int_0^1f(x)dx$ equals $I_0$.
Refer here. I suspect that ...
2
votes
0answers
37 views
Solve $\int _{|z-1|=1} \frac{\log{z}}{z^{4}} dz $
this problem came in my exam today and I couldn't solve it.
This is my failed attempt, using Cauchy's Integral Formula for derivatives.
I think that the theorem tells me that if $z_ {0} \in $ my curve ...
0
votes
2answers
40 views
Is $\lim_{t\to\infty} e^{-at} \int_0^t e^{a t'} f(t') dt'$ ($a>0$) zero?
Given that $f(t)$ is a well-behaved positive function, $f(0)=1$ and $f(t)$ monotonically decreases towards zero as $t\to\infty$.
Can one derive the limit of the expression? I "feel" the ...
1
vote
1answer
51 views
Simplifying the integral $\int \frac{x^2-1}{x^2+1}\cdot \frac{1}{\sqrt{1+x^4}}dx$.
I have the following integral that I must simplify by making two substitutions
\begin{align}
\int \frac{x^2-1}{x^2+1}\cdot \frac{1}{\sqrt{1+x^4}}dx,
\end{align}
one of them is
\begin{align}
y = \frac{...
1
vote
2answers
36 views
Function $f$ with $|f|$ is Lebesgue integrable but $f$ isn't locally Lebesgue integrable
I'm trying to find a function $f$ with $|f|$ is Lebesgue integrable but $f$ isn't locally Lebesgue integrable.
My approach:
Let $X = [0,1]$ and $A \subset X$ with A is the Vitali-set. So $A$ is not ...
1
vote
4answers
92 views
Asymptotic behaviour of $\int_{0}^{1}f(x)x^ndx $ as $n\to \infty $
I'd like to determine the asymptotic behaviour of $$\int_0^1 f(x)x^n\,dx $$ as A) $n\to \infty $ with $0<f(x)\in \mathrm{C}^1([0,1]), 0\le x\le 1$
B) $n\to \infty $ with $0<f(x)\in \mathrm{C}^0([...
-1
votes
0answers
18 views
Γ(z)=M{e^-x}(z) How to write it expanded form? Whether M{e^-x} is constant or complex number when z is complex number. [closed]
Γ(z)=M{e^-x}(z) How to write it expanded form?
Whether M{e^-x} is constant or complex number
when z is complex number.
Γ(z)= ( z-1)! = a complex number result arrived always.
Example (z -1) ! =. ( a +...
1
vote
0answers
21 views
How to minimize some integral by choosing an optimal density function?
The problem I would like to solve is as follows:
$$
\min_{\substack{\hat{f} \\ \hat{f} \geq 0\\ \int\hat{f} \;dw = 1}} \int_{\mathcal{Z}} \left( \int_{\mathcal{W}}\left[\hat{g}(z|w)\hat{f}(w) - g(z|w)...
1
vote
0answers
31 views
Help with Integral Involving fractions and exponentials
Hi I need help with the integral below. I have had a tried a few substitutions but they don't work.
$\int_{z_u}^{z_f} \frac{dz}{\sqrt{\frac{5+\gamma e^{-bz}}{\delta^2}-6}}$
where $\delta$, $\gamma$ $a$...
1
vote
0answers
72 views
Is there an antiderivative for $e^{-\left( x + \frac{1}{x}\right)}$?
I was playing around with some integrals I made up myself and was trying to find a closed-form for
$$
\int_{0}^{t} e^{-\left( x + \frac{1}{x}\right)} \ dx, \qquad t < \infty
$$
I'm aware that if ...
0
votes
1answer
47 views
for what values of parameters p,q does the integral $\int_1^{\infty} \frac{1}{x^p + x^q}dx$ converges?
What I thought is to break this into different cases:
$p=0$: so the integral given ~ $\displaystyle \int_1^{\infty}\frac{dx}{x^q}$ , and it converges for $q>1$.
$p<0$: so the integral given ~ $\...
1
vote
1answer
42 views
Summation by Part and Integration by part.
How is summation by part similar to integration by parts? How are they different?
I was doing some questions on integration by parts and summation by parts and now i am curious to know the ...
1
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0answers
37 views
why is the geometric interpretation for integral $\int_c^d \frac{1}{\sqrt{(x-a)(x-b)}}$ a degree?
why is the geometric interpretation for integral $\int_c^d \frac{1}{\sqrt{(x-a)(x-b)}}$ a degree?
I once saw that an intuitive geometric interpretation is as follows:
This integral reflects the ...
1
vote
0answers
32 views
Average value of $\sqrt{s^2-x^2-y^2}$
This is a function $f(x,y)=\sqrt{s^2-x^2-y^2}$. I want to show that the average value of this function is $\dfrac{2}{3}s$ on the region $D=\{(x,y):x^2+y^2\leq s^2\}$, $s \geq 0$. Assuming that $f(x,y)$...
1
vote
2answers
63 views
Using Leibniz's Rule to Evaluate Integrals
Problem Evaluate the integral
$$\int_0^1 \frac{\ln(x+1)}{x^2+1} \, dx$$
using Leibniz's rule.
Solution attempt.
To evaluate the above integral, we consider a similar integral
$$F(y) = \int_0^1 \frac{\...
1
vote
0answers
34 views
+100
Bounding the propagation function
For $\alpha \in (0,1)$, set $\omega:\mathbb{R^+}\times \mathbb{R^+} \to \mathbb{R}$ defined as following
$$\omega(t;\tau):=1-\pi^{-1}\int_0^\infty \frac{e^{-rt-\tau r^\alpha \cos(\alpha \pi)}\sin(\tau ...
-1
votes
1answer
92 views
Techniques for approximating $\int_{-\pi}^\pi \left(\frac{\sin x}{x}\right)^{30} dx$ [closed]
What are the techniques to approximate the value of this integral?
$$\int_{-\pi}^\pi \left(\frac{\sin x}{x}\right)^{30} dx$$
I have no clue how to go about it.
1
vote
1answer
82 views
Theorem 6.12(a) Of Baby Rudin. Alternative Proof Of $ \int_a^b \left( f_1 + f_2 \right) d \alpha = \int_a^b f_1 d \alpha + \int_a^b f_2 d \alpha$
If $f_1 \in \mathscr{R}(\alpha)$ and $f_2 \in \mathscr{R}(\alpha)$, then
$$ \int_a^b \left( f_1 + f_2 \right) d \alpha = \int_a^b f_1 d \alpha + \int_a^b f_2 d \alpha$$
I want to prove this theorem ...
1
vote
0answers
26 views
Is there a proof for this Theorem related to distributions and almost everywhere equality to 0?
I came across this theorem in one of my courses on Sobolev spaces. I am struggling to find a clear proof for it and I have not succeeded yet.
Some notation first:
$L_{loc}^{1}\left( \Omega \right) $ ...
0
votes
1answer
49 views
Definite Integration With Trig Functions and Euler's Number
I'm trying to calculate a welfare function as a definite integral, and the part I'm having trouble with is the consumer curve function. In econ, this is done as
$\text{CS}=\int_{0}^{Q^{\ast}} \left[f \...
4
votes
0answers
48 views
Show Riemann-integrability of $h:=\max\{f,g\}$
Let be $f,g$ two Riemann-integrable functions on the interval $[a,b]$ where $a<b$. Show that $h:=\max\{f,g\}$ is also Riemann-integrable on $[a,b]$.
My approach:
First, a bit of notation:
\begin{...
0
votes
1answer
47 views
Do we still add $C$ when integrating and the result is an arbitrary function?
I apologize if this question sounds stupid.
I'm just bothered whether I should write
$$u(x,y) = f(x) + C$$
or just
$$u(x,y) = f(x)$$
For the solution of this problem:
$$\frac{\partial u(x,y)}{\partial ...
1
vote
1answer
129 views
Convergence of $\int_0^\infty\frac{\sin x}{x}dx$
I'm working on proving the convergence of the below integral, the value of which I know to be $\displaystyle \frac{\pi}{2}$.
$$
\int_0^\infty\frac{\sin x}{x}dx
$$
I'm also well aware that this ...
0
votes
0answers
37 views
First semester in University Integral Inequality Question [closed]
I have problems with proving the following inequality, which is the last exercise of the chapter of our University book regarding measuring length, area and volume with integrals.
$$ \sqrt{(b-a)^2+(f(...
5
votes
1answer
53 views
Theorem 10.2 Rudin
$\mathscr b(X)$ denotes the set of all complex-valued, continuous, bounded functions with domain $X$.
I don't understand why is $L(h)$ equal of $\prod_{i=1}^k $ $\int_{a_i}^{b_i} h_i(x_i)dx_i$ and ...
2
votes
0answers
35 views
Baby Rudin 10.1
I want to be sure that I understand it correctly.
As I understand it $\int_{I^k} f(x) dx $ = $\int_{a_{k-1}}^{b_{k-1}}$ ($\int_{a_k}^{b_k} f_k(x_1,...x_{k-1},x_k)dx_k$)$dx_{k-1}$ and to continue this ...
1
vote
0answers
50 views
Evaluate $I=\int_{0}^{1}\operatorname{Li}_2^2(x) \operatorname{Li}_3^2(-x)\ln(1+x)\text{d}x$
Denote
$$
\omega_0(t)=\frac{\text{d}t}{t} ,
\omega_1(t)=\frac{\text{d}t}{1-t},
\omega_2(t)=\frac{\text{d}t}{1+t}.
$$
I want to compute($\text{Li}$ denotes polylogarithms)
$$I=\int_{0}^{1}\operatorname{...
2
votes
0answers
60 views
Help proving that signed volume of n-parallelepiped is multilinear
Overview
I am trying to build some intuition about the volumes of parallelepipeds and determinants. I would like to define the determinant as the unique function of $N$ vectors in $\mathbb{R}^N$ which ...
0
votes
1answer
69 views
Evaluate $\iint_D xy$ [closed]
My question is, evaluate the area under the curve, $\iint_D xy$, where $D$ is a region bounded by $y=\sin(x)$ and $y=\cos(x)$. Here is an illustration of it:
As you can see, $D$ is the area between ...
0
votes
1answer
115 views
Where I made mistake(s) as the integrating of $~\int_{}^{}\frac{1}{\sin^{2}\left(x\right)}\,dx~$?
$$L:=\int_{}^{}\frac{1}{\sin^{2}\left(x\right)}\,dx\tag{1}$$
$$=\int_{}^{}\sin^{-2}\left(x\right)\,dx\tag{2}$$
$$=\frac{\sin\left(x\right)^{-1}}{\left(-1\right)\left(\cos^{}\left(x\right)\right)}+\...
0
votes
1answer
29 views
I need the expression of a function which goes to infinity as x goes to 0 and has a finite integral
Is there a function similar to the one in the image that has a discontinuity in the origin, for which:
$$\lim_{x \to \pm\infty} f(x)=0$$
and
$$\lim_{x \to 0^{\pm}} f(x)=+\infty$$
and
$$\int_{-\infty}^{...
