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 ...

learn more… | top users | synonyms (3)

2
votes
0answers
52 views

Time to buy a house without a mortgage equation!!

I am looking into a "real world" calculation to calculate the time taken for someone to buy their own home while they rent it. They do this by buying small pieces of the property every month, and ...
2
votes
1answer
66 views

On convergence a.e and convergence measure

I have a question. First, I know that convergence in measure of a sequence of functions $f_n$ is different than convergence a.e., wich means there are sequences that converge in measure but not a.e. ...
1
vote
0answers
78 views

Prove $f(x)=x$ is Lebesgue integrable on $[0,1]$

Prove that $f(x)=x$ is Lebesgue integrable on $[0,1]$. My definition of integrable comes from Royden's Real Analysis (4th ed). So $f$ is integrable if the lower integral is equal to the upper ...
1
vote
1answer
29 views

Error function - Not seeming to come out right

I have reached two integrals: $$\int_{(x+L)/(2c\sqrt{t})}^\infty e^{-z^2} dz + \int_{-\infty}^{(x-L)/(2c\sqrt{t})} e^{-z^2} dz$$ Now the first evaluates just fine to ...
2
votes
3answers
87 views

Solve y''' = y with given conditions?

I'm given the differential equation: $$y''' = y$$ which solves to: $$y(x) = c_1e^x + e^{-x/2}\left(c_2\cos\left(\frac{\sqrt{3}x}{2}\right) + c_3\sin\left(\frac{\sqrt{3}x}{2}\right)\right)$$ But I'm ...
3
votes
1answer
54 views

$\int_{-\infty}^{+\infty}dx\frac{x\cos(xt)}{e^{ax}-e^{-ax}}$

Apparently from Mathematica we have: $$\int_{-\infty}^{+\infty}dx\frac{x\cos(xt)}{e^{ax}-e^{-ax}}=\frac{\pi^2\mathrm{sech}^2\left(\frac{\pi t}{2a}\right)}{4a^2}$$ for $a,t$ both real and positive. I ...
2
votes
1answer
113 views

Setting up integrals for the moments and the center of mass of a planar region

Problem Consider the following region: a semi-circle with radius = 3 ft on top of a rectangle with height = 11. (with constant density) a.) Set up integrals for the moments, Mx, My, and the center ...
3
votes
3answers
262 views

Finding the Integral of a function

$\int \dfrac{1}{9+x^2}dx$ The answer is $\arctan(x/3)/3 + C$ , but I don't understand the process of how the answer was found. I tried using u-substitution, but I came up with 2xdx/x^2+9.
16
votes
3answers
395 views

Closed-form of $\int_0^1 \operatorname{Li}_3^3(x)\,dx$ and $\int_0^1 \operatorname{Li}_3^4(x)\,dx$

We know a closed-form of the first two powers of the integral of trilogarithm function between $0$ and $1$. From the result here we know that $$I_1=\int_0^1 \operatorname{Li}_3(x)\,dx = ...
0
votes
1answer
60 views

Evaluating the integral of $1+z+1/\tan z$ over a circle

I am a beginner and I want to learn how to solve these kind of integrals: $$\int_{|z|= \pi/4}\left(1+z+\frac{1}{\tan z}\right)\,dz$$ So should I divide it in three integrals, calculate each integral ...
0
votes
1answer
25 views

Solving a separable differential equation: What's wrong with my calculation?

Solve the following separable differential equation:$$\frac{dy}{dx}=(-4)\cdot e^y \cdot cos(4x)$$ My answer (which is incorrect but I don't know why): $$\frac{dy}{dx}=(-4)\cdot e^y \cdot cos(4x)$$ ...
0
votes
0answers
47 views

Is this integral in its most simplified form?

The following integration $$F(x)= \int_{x}^{+\infty} \frac{t}{1+t^\alpha} dt$$ cannot be solved in general, however can be expressed when $\alpha=4$ as $$F(x)= 0.5 \text{tan}^{-1} (x^{-2}) $$ it can ...
3
votes
2answers
407 views

Calculating integral of function with square root

I don't have any idea how to calculate this integral. I learned to calculate the elementary functions, and how to calculate by the positioning way. For example $t=e^x+1 \rightarrow dt=e^x \, dx$ and ...
3
votes
1answer
62 views

Is this an acceptable way to integrate?

I am supposed to find: $$ \int \sec(1-x)\tan(1-x) dx $$ I then set $ u = \sec(1-x) $ $$ du = -\tan(1-x)\sec(1-x)\ dx $$ therefore $$ \frac{-du}{\sec(1-x)} = \tan(1-x)\ dx$$ Which when applied gives ...
2
votes
1answer
34 views

Integration of $4xe^{-1/2}$

It is given that $$\int_0^p4xe^{-\frac{1}{2}x}dx=9$$ where $p$ is a positive constant (i) Show that $$p=2 \ln \left( \frac{8p+16}{7} \right )$$ I reached $$8pe^{-p/2} + 16e^{-p/2} = 7$$ What are ...
0
votes
1answer
27 views

asymptotic series for “stable distribution”

I'm trying to understand how to get from one equation to another in a certain paper I am studying (DOI:10.1080/00018738100101467, eqs. 4.34 and 4.35). The equations are pretty self contained, so I'm ...
1
vote
2answers
125 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
61 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
19 views

Linear functionals and integration verification

Can you please verify my reasoning? (a) Yes as (b) No, as function is squared (c) Yes, same reasoning as (a), squared values of x do not affect linearity. Does the region of integration affect ...
1
vote
1answer
99 views

Find the area bounded by $x+y=3$ and the coordinate axes.

Find the area bounded by $x+y=3$ and the coordinate axes. I know how to find the area bounded by 2 curves it's just that I'm confused with "coordinate axes". Is it the same as x=y? or not? please ...
0
votes
0answers
28 views

Existence and uniqueness of Volterra integral equations of the first kind with vanished kernel [duplicate]

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To prove the existence and uniqueness of solutions of Volterra integral equation(VIE) of the first kind, we usually differentiate it and convert to the VIE of the ...
11
votes
2answers
156 views

How does one evaluate $\int \frac{\sin(x)}{\sin(5x)} \ dx$

The below problem is taken from Joseph Edwards book Integral Calculus for beginners. How does one show: $$5 \int \frac{\sin(x)}{\sin(5x)} \ dx= \sin\left(\frac{2\pi}{5}\right) \cdot ...
0
votes
1answer
46 views

Darboux integrals with bisected partition

Let us call $\overline{\int_a^b}f(x)dx$ the Darboux upper integral of $f$ and $\underline{\int_a^b}f(x)dx$ the lower one. Let us construct a partition of $[a,b]$ into $2^n$ intervals $[x_{k-1},x_k]$ ...
2
votes
1answer
44 views

Differentiability of the convolution $\int_0^tf(t-s)g(s)\;ds$

Given two continuously differentiable functions $f,g:[0,\infty)\to\mathbb{R}$. I want to know what we can tell about the differentiability of $$(f\ast g)(t)=\int_0^tf(t-s)g(s)\;ds$$ Especially, why ...
5
votes
2answers
128 views

Is this closed-form of $\int_0^1 \operatorname{Li}_3^2(x)\,dx$ correct?

According to Freitas' paper at page $11$. $$\int_0^1 \operatorname{Li}_3^2(x)\,dx = 20-8\zeta(2)-10\zeta(3)-\frac{15}{2}\zeta(4)-2\zeta(2)\zeta(3)+\zeta^2(3).$$ I evaluated the LHS and it is ...
0
votes
2answers
36 views

Calculating a calculable Integral using Integration

I am having trouble with an integral: $$\frac2L\int_L^\infty C\sin\left(\frac{n\pi x}{L}\right) dx$$ Where $C$ is just a constant. I can't see how to do this, despite it apparently being rather ...
5
votes
3answers
117 views

Proving convergence of $ \int \limits_0^{\infty} \cos\left(x^2\right) dx $

How would one prove the convergence of $$ \int_0^{\infty} \cos\left(x^2\right) \,\mathrm dx $$ I tried using the integral test for convergence by noting that making the substitution $u = x^2$ means ...
0
votes
0answers
12 views

find the kernel of voltera 2nd kind with particular form 2. (Alternative approach)

find the kernel of voltera 2nd kind with particular form 2. (Alternative approach) in which we take kernel function of x and t ot just x or just function of t. we try to solve it with alternative ...
5
votes
4answers
363 views

Some integral representations of the Euler–Mascheroni constant

What kind of substitution should I use to obtain the following integrals? $$\begin{align} \int_0^1 \ln \ln \left(\frac{1}{x}\right)\,dx &=\int_0^\infty e^{-x} \ln x\,dx\tag1\\ &=\int_0^\infty ...
2
votes
6answers
220 views

How to solve $y''' = y$

I'm trying to solve the following differential equation $ y''' = y$ and given conditions: $ y(1) = 3$, $y'(1) = 2$ and $y''(1) = 1 $ I began by making it: ...
3
votes
3answers
59 views

Using $x=\tan \theta$ to solve $\int x\sqrt{1+x^2}\,\mathrm dx$

I'm having a lot, I repeat, a lot of trouble with Calculus II, particularly trigonometric substitution. At the moment, I'm extremely confused as to how to integrate $\int x\sqrt{1+x^2}\,\mathrm dx$ ...
0
votes
0answers
13 views

Converting $\sum_n\cos(\omega_nt)G(\omega_n)$ for arbitrary $G$ into an integral over $\omega$

In Gardiner's 'Quantum Noise' he considers some function $f$ defined by: $$f(t)=\sum_n\cos(\omega_nt)G(\omega_n)$$ He then lets the spectrum $\omega_n$ approaches a continuum, in which case: ...
9
votes
1answer
391 views

Log integrals IV

It can be determined that the integral \begin{align} \int_{0}^{\pi/2} \frac{x}{\sin(x)} \ln\left(\frac{1+\cos(x) - \sin(x)}{1+\cos(x) + \sin(x)} \right) dx \end{align} has a finite value. Is there an ...
13
votes
1answer
221 views

Integral ${\large\int}_0^1\frac{\ln^2\ln\left(\frac1x\right)}{1+x+x^2}dx$

Gradshteyn & Ryzhik, 7th ed., p. 570, formula 4.325(5) give the following definite integral: ...
0
votes
1answer
79 views

How can the integral of $|\sin(x)|$ be $-\cos(x)\text{sgn}(\sin(x))$?

Wolfram|Alpha tells me that $\int|\sin(x)| = -\cos(x)\text{sgn}(\sin(x))$ (which happens to also be its derivative), but I don't understand how this is possible, because the resulting function jumps ...
1
vote
1answer
60 views

Smoothing Lemma

Given a $C^0$ function $g:[a,b]\to \mathbb{R}$ that is smooth everywhere except at $c$ (where $a<c<b$), and has positive derivative everywhere except at $c$, the claim is that there exists a ...
0
votes
1answer
30 views

Solve the integral with complex number and floor function.

let $z\in\mathbb{C},\,0<\left|z\right|<1$.I would like to calculate the integral $$\int_{1}^{\infty}\frac{\left(1-z\right)^{\left\lfloor t\right\rfloor }}{t^{2}}dt$$ where ${\left\lfloor ...
1
vote
2answers
41 views

Area between $y=x^4$ and $y=x$

The problem I'm having some trouble solving is this: calculate the area between $y=x^4$ and $y=x$. The points are $a = 0$ and $b = 1$, but the definite integral is negative. What am I doing wrong ...
0
votes
1answer
50 views

Integral for $\frac{x}{x^2+1}cosx$

When computing Fourier transformation I came across these integral: $$ \int_{\Bbb R}\frac{x \cos x}{1+x^2}\;dx\text{ or } \int_{\Bbb R}\frac{x \sin x}{1+x^2}\;dx $$ Can anyone give me some hints on ...
1
vote
0answers
33 views

Looking for a reference of integral involving product of four spherical harmonics

We know $$\int d \Omega Y_{l_1m_1}(\theta,\phi) Y_{l_2 m_2}(\theta,\phi) Y_{l_3 m_3 } (\theta,\phi) = \sqrt{ \frac{ (2l_1 + 1)(2 l_2+1)(2l_3+1)}{4\pi} } \pmatrix{ l_1 l_2 l_3 \\ m_1 m_2 m_3 } ...
0
votes
3answers
36 views

Semi Gauss integral limit

I am courrently stuck at showing that: $lim_{x \rightarrow \infty}\int_0^xe^{t^2-x^2}dt=0$. Non of my tries by estimations lead to succes so I would appriciate any kind of help.
3
votes
0answers
57 views

How to evaluate the following integrals

$$\int\limits_0^{\frac{\pi }{2}} {{x^2}{{\ln }^2}\left( {\sin x} \right)\ln \left( {\cos x} \right)dx} ,\int\limits_0^{\frac{\pi }{2}} {x\ln \left( {\sin x} \right){{\ln }^2}\left( {\cos x} \right)dx} ...
2
votes
2answers
37 views

How to prove this multivariable integral identity?

By numerical experimentation I found that $$ \lim_{\beta \rightarrow \infty} \frac 1 \beta \int_0^{\beta}dx \int_0^{\beta}dy \, f\left( |x-y| \right) = 2\int_0^{\infty}dx \, f(x) $$ if $f:\mathbb{R} ...
0
votes
1answer
106 views

calculus integration, average height of point on semi circle

i was recently watching a single variable calculus video of mit 18.01, lecture 23. in that it is said that average height of a point on semicircle with respect to arc length is 2/pi.I have a hard time ...
0
votes
1answer
47 views

Name for LDC: Lebesgue?

Is there also a name associated to the Lebesgue dominated convergence theorem like Beppo-Levi or Fatou? Would Lebesgue be reasonable? Who did originally prove it?
1
vote
1answer
46 views

Is this map surjective?

Let $B^1(\mathbb{R},\mathbb{R})$ be the set of all locally integrable functions $f:\mathbb{R}\to \mathbb{R}$ such that $$\sup_{t\in \mathbb{R}} \int_t^{t+1}|f(x)|dx<\infty.$$ Consider the map ...
0
votes
2answers
75 views

Substation problem with a simple integral

i have this integral $$ \int {4x\over \sqrt{1+4x^2}} dx $$ and i have tried to solve it by doing like this $$ t=\sqrt{1+4x^2}->t^2=1+4x^2->2tdt=8xdx->tdt=4xdx $$ and im gettin this integral ...
0
votes
4answers
147 views

Why isn't area under curve from 0 to infinity of $\frac{1}{x^2}$ equal to 3?

The integral from $$\int_1^\infty \frac1{x^2} \ dx = 1$$. The area of the box bounded by $x = [0,1]$ and $y = [0,1]$ is $1$. For the area between $x = [0,1]$ and $y = [1,\infty)$, consider the area ...
1
vote
1answer
59 views

Measurability and a integral

I need to calculate $\lim_{n\rightarrow\infty}\int^{\infty}_{0}\frac{cos(\frac{x}{n})}{2^x}d\lambda(x)$ and show that the integral makes sense for every $n$. My approach so far: Let ...
1
vote
1answer
53 views

If $\mathbb{E}(X^\alpha)<\infty$ for $0<\alpha<1$ show that $\mathbb{E}(\min (X,t))$ is $o(t^{1-\alpha})$

If $X\geq 0$, and $\mathbb{E}(X^\alpha)<\infty$ for $0<\alpha<1$ show that $\mathbb{E}(\min (X,t))$ is $o(t^{1-\alpha})$. We know that that $$\mathbb{E}(\min(X,t))=\int_{X\leq ...