1
vote
2answers
93 views

Factorial Rational Limit

Anything besides the squeeze theorem. Here it is: $$\lim_{n\to\infty} \frac{(2n - 1)!}{{2n}^{n}}$$ Can someone start me off?
2
votes
1answer
20 views

Finding Factorial using Integral Definition

$n! = \int_{0}^{\infty} {e}^{-x}{x}^{n} \,dx$ How can we find $400!$? $400! = \int_{0}^{\infty} {e}^{-x}{x}^{400} \,dx$ Integration by parts is way too complicated, what are the other options?
0
votes
1answer
38 views

What can be said about the inverse of the antiderivative of a strictly positive function?

Let $f:\mathbb R\rightarrow [1,\infty)$ be (a strictly positive) function. Define $$F(t) = \int_0^t f(s)ds.$$ Obviously, $F$ is injective and hence invertible. How does $F^{-1}$ look like?
1
vote
0answers
6 views

Mathematical literature for Dirichlet-multinomial integration

I am playing with topic models (Latent Dirichlet Alocation + modifications) and would be grateful for pointers to mathematical analysis literature that covers details of the derivations. For ...
0
votes
2answers
95 views

Show that $\,a_n=f(1)+f(2)+\cdots+f(n)-\int_1^n f(x)\,dx\,\,$ converges

Let $\,f:[1,\infty)\to \mathbb R\,$ be a decreasing and lower bounded function. Show that the sequence $\{a_{n}\}_{n\in\mathbb N}$ defined as: $$ a_n=f(1)+f(2)+\cdots+f(n)-\!\int_1^n\!\! f(x)\,dx, $$ ...
0
votes
2answers
99 views

Extremely tough indefinite integral

This integral does indeed use special functions, so do include them here. Evaluate: $\int \frac{1}{\sqrt{x}\ln(x)} dx$ $x = {\sqrt{x}}^{2} \space \text{let} \space u = \sqrt{x}$ $= 2\int ...
1
vote
2answers
60 views

Evaluate $\int_{\partial C} \frac{dz}{(z-a)(z-b)}$ where $\partial C$ is the boundary of a rectangle ($a$ and $b$ are not on $\partial C$)

In discussing the possible outcomes of the integral $$\int_{\partial C} \frac{dz}{(z-a)(z-b)}$$ where $\partial C$ is the boundary of a rectangle ($a$ and $b$ are complex and not on $\partial C$), ...
2
votes
2answers
66 views

I would like prove a result in integration

I would like prove this result $$\int_0^1 \frac{\left(\log (1+x)\right)^2}{x}\mathrm dx=\frac{\zeta(3)}{4}$$
9
votes
3answers
295 views

Evaluate $ \int_0^\pi \left( \frac{2 + 2\cos (x) - \cos((k-1)x) - 2\cos (kx) - \cos((k+1)x)}{1-\cos (2x)}\right) \mathrm{d}x $

Evaluate the following definite integral: $$ \int_0^\pi \left( \frac{2 + 2\cos (x) - \cos((k-1)x) - 2\cos (kx) - \cos((k+1)x)}{1-\cos(2x)}\right) \mathrm{d}x, $$ where $k \in \mathbb{N}_{>0}$.
4
votes
0answers
247 views

Explain this step in lecture notes

The bounty offered is for the person that explains me how the author gets from equation 3.19 to equation 3.20 in these lecture see here. Normally I would agree that copying the relevant equation would ...
3
votes
1answer
111 views

Books with collections of unusual and advanced integration techniques

I am searching for some comprehensive books that collect, explain, and provide examples of extremely advanced and/or unusual integration techniques. Can you point out some good references? Note: ...
0
votes
0answers
30 views

Proving inequality with complex Riemann-Stieltjes Integral

I am trying to prove the following proposition (IV.1.17b) from Conway's Functions of One Complex Variable I: Let $\gamma$ be a rectifiable curve and suppose that $f$ is a function continuous on $\{ ...
1
vote
2answers
85 views

$\int_{23\pi}^{71\pi/2}\ln \left ( \frac{\left ( 1+\sin x \right )^{1+\cos x}}{1+\cos x} \right )\,dx$

I ran into this integral and I'm trying to solve it. $$I=\int_{23\pi}^{71\pi/2}\ln \left ( \frac{\left ( 1+\sin x \right )^{1+\cos x}}{1+\cos x} \right )\,dx$$ Well, this has something to do with ...
1
vote
0answers
32 views

Liouville's Extension of Dirichlet Theorem

Can we use Liouville's Extension of Dirichlet Theorem to find triple integral $\int\int\int(u^2+v^2+w^2)\space du\space dv\space dw\space where\space u=0, v=0, w=0\space \&\space u+v+w\leq1$? Or ...
0
votes
0answers
22 views

Integrating a particular function.

Could someone please show me how to integrate the following: $\int_{-\infty}^{\infty}p(x)L(p(x))dx$, where $L(x)$ is defined as $L(x)=\frac{x-1}{ln(x)}$ and $p(x)=(2\pi ...
0
votes
1answer
77 views

About Riemann integrability

I need to prove if $f$ is continuous on an interval $I$, then its Riemann integral exists. It is hard for me because it is an interval and not closed interval. Can anyone give me some answers or ...
1
vote
1answer
55 views

Autonomous differential equation

Let $f: \Bbb R \to \Bbb R$ and $x_0 \in \Bbb R$, such that $f(x_0)> 0 $, and assume that $x(t)$ is the solution of $x'=f(x)$, such that $x(0)=x_0$. If $f(x) > 0$ then $x(t)$ is defined for all ...
0
votes
2answers
90 views

How to write this integral in a nice way?

I have a function $f(a,b):= \int_{-1}^{1} e^{i (ax+bx^2)}dx$ with $(a,b) \in \mathbb{R}^2 \backslash \{(0,0)\}$ and now I want to find out what $|f(a,b)|^2$ is. Is there a way to write this in a ...
2
votes
1answer
41 views

$\int_0^{2 \pi} \cos(x)e^{i (a \cos(x) + b \cos^2(x)} dx$ and $\int_0^{2 \pi} \cos^2(x)e^{i (a \cos(x) + b \cos^2(x)} dx$

I am currently dealing with the two integrals in the title and I want to find out, when their real part of their imaginary part vanishes ( so for which constellation of $(a,b) \in \mathbb{R}^2 ...
1
vote
0answers
17 views

What assumptions should be made?

take a problem like A trough is 12 feet long and 3 feet across. Its ends are isosceles triangles with altitudes of 3 feet. Water is being pumped into the trough at 2 cubic feet per minute. How fast ...
1
vote
1answer
32 views

Why $f (x):= \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)}$ only belongs to $L^2(0, \infty)$

This is a result given in Royden and Fitzpatrick (p. 143). Show that $$ \int_0^\infty \left[ \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)} \right]^p < \infty $$ if and only if $p=2$. That ...
0
votes
0answers
53 views

How to find if an integral is possible to compute: Failing to solve integral for quadratic functional

I am trying to solve the below integral, and no computational method seems to be capable of solving this, nor can I do it by hand. Any ideas? $$\int_{t_0}^{t_1}[a(t)((2\dot{x^*}\dot{\eta} + ...
0
votes
1answer
34 views

Proving the integral of a discontinuous function

Let $y_n$ be a monotone decreasing sequence with $\lim_{n\to\infty}y_n=0$. Define the function $f:\left[0,1\right]\to\mathbb{R}$ by $$ f(x)= \begin{cases} y_n &\text {there ...
0
votes
0answers
70 views

Does this integral have a closed form?

$I=\int_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$ I saw this question here. It is really hard to find a closed form. Or is there no closed form? Please give me a hand. Thanks!
5
votes
2answers
94 views

Green's function for $y''+y=f(x)$

This example is taken from the Wikipedia's article. Namely, find the Green's function for $$y'' + y = f(x)$$ with boundary conditions: $$y(0) = y(\frac {\pi} {2}) = 0.$$ The defining equation for ...
0
votes
1answer
47 views

When does this integral converge?

So I'd like to find out for which values of $a,b>0$ the following integral is well-defined and how that will change if the absolute value is removed? Thanks! $I = \int_0^\infty ...
0
votes
1answer
37 views

Proving Riemann integrability. [closed]

Let $y_n$ be a monotone decreasing sequence with $\lim_{n\to\infty}y_n=0$. Define the function $f:\left[0,1\right]\to\mathbb{R}$ by $$ f(x)= \begin{cases} y_n &\text {there ...
4
votes
2answers
88 views

What is wrong with the following u-substitution?

We will calculate $\displaystyle\int^{2 \pi}_0 x \, dx$. Let $u=\sin (x)$, and observe that $\sin(2 \pi)=0$ and $\sin(0)=0$. We also have that $\frac{du}{dx}=\cos(x)=\sqrt{1-u^2}$. Hence, $$ \int^{2 ...
1
vote
1answer
34 views

A certain relation of a polynomial to its coefficients

I've got a certain problem: If $A(t) = a_0+a_1t+ ...+a_Nt^N$, show that: $a_k = \frac{1}{2\pi}\int_{-\pi}^{\pi} e^{-ikx}A(e^{ix})dx$ after some rearrangements I got: $a_k = ...
0
votes
0answers
36 views

Integration of a differential equation

I've got some problems with integrating a ODE, so maybe someone could add some words of advice. Given the following equation: $z''(x)-2\gamma z'(x) +p(x)z(x)=0$, $(1)$ and $\varphi(x)=z(x)e^{-\gamma ...
1
vote
3answers
44 views

triple integral and limits

$$\iiint\limits_H (x^2+y^2) \, dx \, dy \, dz \\ H=\{(x,y,z) \in R^3: 1 \le x^2+y^2+z^2 \le9, z \le 0 \}$$ I'm using Spherical coordinate system: $$x=r\cos \theta \cos\phi $$ $$y=r\cos \theta ...
0
votes
1answer
25 views

Calculate double integral on the specific field

Calculate integral $\int\int_D \frac{dxdy}{(x^2+y^2)^{3/2}}$ $ D=\{(x,y)\in R^{2} : x^2+y^2 \le1, -x \le y, x+1 \le y\} $ I draw a graph and this is a little part of a disk. As I calculate $x \in ...
5
votes
0answers
61 views

How prove $f(a_{i})=0$ if $\int_{0}^{1}x^kf(x)dx=0,k=1,2,3,\cdots,n$ [duplicate]

let $f(x)$ is continuous on $[a,b]$,and such $$\int_{0}^{1}x^kf(x)dx=0,k=0,1,2,3,\cdots,n$$ show that: there exsit $n+1$ different $a_{1},a_{2},\cdots,a_{n},a_{n+1}(a_{i}\neq a_{j},\forall ...
0
votes
1answer
25 views

Laplace transform quick answer check :) using second shift theorem

I want to get $L((t-4)^2u(t-4))$ I say this is a second shift with $g(t)=(t^2-4t)$ and my friend says "NO you are wrong, you are dumb!!!!!! $g(t)$ is MOST CERTAINLY equal to $t^2$" Mine gives me ...
0
votes
1answer
20 views

line integrals and partial derivatives statement (Green's theorem application)

Let $P(x,y),Q(x,y)$ be $C^1$ functions of $\mathbb R^2$, prove that the following statements are equivalent: (1) $P_x-Q_y=0$ and $P_y+Q_x=0$ (2) For every simple closed curve $C$, it is satisfied ...
0
votes
0answers
32 views

estimation of an integration?

How to integrate the following expression, given $\int \rho(x)dx=M>0$, $\rho\geq 0$, and $\omega$ is a function? \begin{equation} \int \frac{x_1y_2-x_2y_1}{2\pi|x-y|^2}\rho(x)\omega(y)dxdy ...
1
vote
1answer
33 views

From $\left\lVert \sup_{M>M_0} \left| \sum_{k=M_0}^M f_k \right| \right\lVert_2 < \epsilon$ show convergence a.e. of the series.

I'm having trouble with the following 'qual' problem. For one, I don't know what to make of the absolute value inside the $L^2$-norm. In short, I just don't have any intuition for it. And I don't ...
1
vote
1answer
36 views

Differentiation under the integral sign (one complex variable)

Let $u(z), u'(z)$ be complex-analytic functions on an open neighborhood $\Omega \subseteq \mathbb{C}$ of the origin. Also, let $f(X)$ be a complex-analytic function. For $s \in [0,1],$ define $$g(s,z) ...
0
votes
0answers
18 views

How to compute using integration the areas of the dodecagons (i.e. twelve-sided polygons) inscribed and circumscribed around a unit circle?

How to compute the areas of the dodecagons (i.e. twelve-sided polygons) inscribed and circumscribed around the unit circle centered at the origin using the methods of the integral calculus?
0
votes
1answer
23 views

What is the value of the unknown parameter so that the given area condition holds?

The graphs of $f(x) \colon= x^2$ and $g(x) \colon= cx^3$, where $c > 0$, intersect at the points $(0,0)$ and $(1/c, 1/c^2)$. What is the value of $c$---and how to compute this value---so that the ...
1
vote
4answers
72 views

How to prove that $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists

I am trying to show that the integral $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists ($n$ is a natural number). I tried to use the comparison theorem by bounding from above the ...
0
votes
1answer
46 views

Square integrability of functions

Suppose that for a function $f(x)\,\,, x\in\mathbb{R}$ holds \begin{align} \int_{0}^{T}|f(x)|^{2} ~\mathrm{d}x<\infty \end{align} Does it also holds that \begin{align} ...
0
votes
1answer
39 views

Arc lenght of a curve is finite

Let $b<0<a$, and consider the function $\alpha:(0,+\infty) \to \mathbb R^2$ defined as $$\alpha(t)=(ae^{bt}\cos(t),ae^{bt}\sin(t))$$ Show that $\lim_{t \to +\infty} \alpha'(t)=(0,0)$ and ...
0
votes
5answers
111 views

Assumptions in Word Problems (Calculus)

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
1
vote
1answer
127 views

How to find this integral $\int_{0}^{\infty}\dfrac{f(x)}{g(x)}dx$ [duplicate]

show that: $$I=\int_{0}^{\infty}\dfrac{x^8-4x^6+9x^4-5x^2+1}{x^{12}-10x^{10}+37x^8-42x^6+26x^4-8x^2+1}dx=\dfrac{\pi}{2}$$ I found this : ...
1
vote
1answer
27 views

Discretization of an integral

Given $f: [a,b] \to R $ and $K: [a,b]$ x $[a,b]$ $\to R$, we want to find a solution $\varphi:[a,b] \to R $ to the Fredholm integral equation: $$\varphi(x) = f(x)+\int _{ a }^{ b }{ K( x,t)\varphi ...
3
votes
1answer
98 views

Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$

I want to solve the following two integrals analytically \begin{aligned} I_1 = & \int\limits_0^{\infty}\frac{e^{a/x^2}}{1-e^{b/x^2}}e^{-x^2}dx \\ I_2 = & ...
0
votes
2answers
103 views

Does This Function Exist?

I am trying to construct a piecewise function $g:[0,1] \rightarrow \mathbb{R}$ with $g(0)=0,\hspace{3mm}$ $g \geq 0$, $\hspace{3mm}$ $\int_0^x g(t)dt\leq x,\hspace{3mm}$ and such that there is a ...
4
votes
3answers
118 views

Solution of $\frac{d^2y}{dx^2} - \frac{H(x) y}{b} = H(-x)$

Does the equation $$\frac{d^2y}{dx^2} - \frac{H(x)}{b} y = c H(x)$$ have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant? Update: What about the second step ...
2
votes
1answer
74 views

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.