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
22 views

Evaluating an integral by dominated convergence theorem

I would like to know how to solve this two problems: a) $$ \lim_{n\to \infty}\int_0^n \left( 1-\frac{x}{n} \right)^{-n}\log{(2+\cos(x/n))} \, dx $$ b) $$ \lim_{n\to \infty}\int_0^{\infty} n e^{-nx} ...
2
votes
4answers
69 views

There is some strategy to solve an integral of this kind?

How to solve the integral $$\int\frac{\ln x}{\sqrt{1-x}}dx$$ and $$\int\sqrt{\frac{x}{x-1}}dx$$ I have no idea of how to deal with these integrals. It's the first integral I attempted. ...
0
votes
1answer
25 views

Integral involving CDF of a normal distribution

Can we evaluate the following integral ? $$\int_0^\infty x e^{-x^2} \Phi(ax+b)\,\mathrm dx$$ Here $\Phi(\cdot)$ is the cumulative probability distribution function of a standard normal random ...
2
votes
2answers
17 views

Finding the bounds of a solid for triple integrals

Ok, so I have an answer, most likely the wrong one. The question being asked is: Using polar coordinates find the volume of the solid bounded below by the $xy–plane$ and above by the surface $x^2 ...
0
votes
0answers
9 views

Integral of a funtion avoiding hypergeometric functions

I'm solving the following differential equation: $$uy''(u)+\gamma y'(u)+\frac{1}{u(1-u)}=0,~~\gamma=constant$$ For that, I transform this equation into a first orer one: $$uf'(u)+\gamma ...
-1
votes
1answer
33 views

What do these questions mean?

Can someone please explain what these questions mean in simpler terms? 1)If g is continuous on $[a,b]$ and for all $ x ∈ [a,b] $ we have $ g(x) ≥ 0 $ and also $ g(x_o) > 0$ for some $x_o ∈ ...
0
votes
0answers
20 views

Divergence theorem to find flux

I am trying to use divergence theorem to find flux out of the solid $x^2+y^2\leq 9, 0\leq z\leq1$, given that $v(x,y,z)=xi+3y^2j+2z^2k$, So my attempt is that i computed the divergence which comes ...
0
votes
2answers
18 views

Separable ODE (with partial fraction integration)

Im stuck half way trying to solve the equation below. I tried using partial fraction integration and I think Im somewhere near. I need to express my answer as stated in the picture as well Thanks in ...
0
votes
1answer
21 views

Finding limits of integration for double integral

Given a region where the $x$ limits are $-1< x<1$ and $0< y<\sqrt{4-x^2}$, with the option of converting into polar coordinates, i.e. the function $(x,y)$ can be replaced by $r^2$. I'm ...
1
vote
1answer
15 views

Finding the volume of the following solid using triple integrals

Find the volume of the solid in the first octant bounded by the coordinate planes, the cylinder $x^2 +y^2 =4$ and the plane $z+y=3$. I found the integral bounds just fine. So I have $\int_{0}^{2} ...
2
votes
1answer
21 views

Evaluating area using an integral in polar coordinates

I am trying to find the area of a circle which is given by the polar parameterization $$r(\phi) = \cos\phi + \sin\phi.$$ I can evaluate it in 2 ways and don't know why I get different answers. First ...
0
votes
0answers
27 views

Finding an appropriate scalar

I have the following integral $$a_N\int_0^\infty{\frac{2e^x}{e^{2x}+T_{N-1}(2x)}}dx=1$$ Where $T_N=\sum_{k=0}^N\frac{x^k}{k!}$ I'd like to find the value for $a$ which makes this true for all ...
0
votes
0answers
26 views

How to convert infinite intergral to sum

How to convert Wiener filter formulas from integral to sum? They are for images therefore it must be possible to convert them to sums. Any help will be appreciated: I could not find much info on ...
0
votes
0answers
24 views

Solve this integral or at least find an upper bound?

Let $r,t>0$ be fixed. Let $a,b,c$ be numbers such that the following integral converges (I think $a,b,c>-1$ is OK). Then I would like to compute the following integral explicitly if possible or ...
0
votes
0answers
22 views

A Integral inequality.

For any positive integer $n \in {\mathbb{N}^ + }$, prove inequality $$\int_{ - \pi }^\pi {\left| {\cos \left( {\frac{{2n + 1}}{2}t} \right)} \right| \cdot \left| {\frac{1}{{\sin \left( {\frac{t}{2} + ...
2
votes
2answers
48 views

A problem in definite integral.

What will be the value of $a$ for which the integral $$\int \limits^{\infty }_{0}\frac{dx}{a^{2}+(x-\frac{1}{x})^{2}} =\frac{\pi}{5050}$$ where $a^{2}\geq0$ It seems like a standard integral but ...
0
votes
1answer
13 views

Integral and dominated convergence theorem

Let us define $g_n(x)= n\chi_{[0,n^{-3}]}(x)$. I am looking for help to answers the following problem $(a)$ Show that if $f$ $\epsilon$ $L^2([0,1])$ then $\int_0^1f(x)g_n(x)dx \rightarrow 0$ as $n ...
0
votes
2answers
29 views

Is the following Differential Equation undefined for given values of X & Y?

I have been presented with the following differential equation which I'm asked to solve, where $y=0$ when $x=\pi$. $$(y+1)\sin x\frac{dy}{dx} = (y^2+1)\tan^2x$$ I notice that $(y^2+1)$ may be ...
1
vote
1answer
28 views

find $\lambda$ such that the integral has a solution.

I have the integral equation: $u(x) = f(x) + \lambda \int_0^{\frac{1}{2}}u(y)dy$ I have to find $\lambda$ such that the integral has a solution. How to approach such problems?
2
votes
0answers
43 views

How to find this integral from Gaussian integral?

How to find $$\int_0^\infty e^{-ax^2-\frac{b}{x^2}}dx$$ using gaussian integral? I tried complete the square: $$-ax^2-\frac{b}{x^2}=-\left(\sqrt{a}x+\frac{\sqrt{b}}{x}\right)^2+2\sqrt{ab}$$, but what ...
-1
votes
5answers
87 views

Where is the mistake in proving 1+2+3+4+… = -1/12?

https://www.youtube.com/watch?v=w-I6XTVZXww#t=30 As I watched the video on YouTube of proving sum of $$1+2+3+4+\cdots= \frac{-1}{12}$$ Even we know that the series does not converge. First I still ...
1
vote
2answers
31 views

Double integral over a parallelogram

I understand the general concept behind double integrals but do not understand how to change the coordinates linearly, and what to do from there. Find $$\int\int_P(x+y)dxdy$$ Where $$P$$is ...
-3
votes
1answer
26 views

Find the indefinite integral and check the result by differentiation

Find the indefinite integral and check the result by differentiation. I have worked all the problems just I am stuck and would like to check my answers. (1) $\int(x-x^2)dx$ (2) ...
0
votes
3answers
18 views

Integration with a variable in the terminals.

I know that in general, if we integrate over some defined values of $x$ and $y$ we find that, for a function $f(x,y)$ $$\iint f(x,y) \ dxdy=\iint f(x,y) \ dydx.$$ However, if we were to integrate, for ...
0
votes
0answers
20 views

Integrating Associated Legendre Polynomials

As part of a derivation for the question I asked here in Physics stackexchange, I am trying to calculate the following integral, but I am not sure how to proceed: ...
2
votes
0answers
45 views

How to integrate $\frac{1}{\sqrt{x^2+y^2+z^2}}$

want to evaluate $$\int\frac{1}{\sqrt{x^2+y^2+z^2}}dxdydz$$ over entire $\mathbb{R}^3$ except $(0,0,0)$. I did this using polar coordinate and got ...
1
vote
2answers
42 views

What does the notation $\int_A$ mean, where $A$ is an event in a probability space?

I am used to seeing integral notation like this, which means the integral over the domain from a to b. $$ \int_{a}^{b} $$ But now I am looking at a statistics book that says "let A be an event" and ...
5
votes
2answers
209 views

Basic integration question.

I have the integral $$\iint x^2y^2 \ dx\,dy$$ but I am meant to evaluate it at the limits $0<y<1$ and $-2y<x<2y$. I am wondering what terminals of integration I should put in for $x$. Do I ...
0
votes
0answers
33 views

How to find the average velocity of blood?

The velocity $v$ of blood that flows in a blood vessel with radius $R$ and length $L$ at a distance $r$ from the central axis is $$v(r) = \frac{P}{4\eta L}(R^2 − r^2)$$ where $P$ is the pressure ...
0
votes
2answers
46 views

Exponential Integration [duplicate]

I don't know how to solve this equation: $$\int_0^\infty e^{-x} (x-a)^m dx$$ where $a$ is a constant and $m$ = $n+1$ Thanks in advance for your help.
3
votes
6answers
52 views

How can I prove that $ \int \text{sech}(x) ~ \mathrm{d}{x} = {\sin^{-1}}(\tanh(x)) + c $?

How can I prove that $$ \int \text{sech}(x) ~ \mathrm{d}{x} = {\sin^{-1}}(\tanh(x)) + c? $$ I don’t know how to prove this identity. Any help? I tried to multiply by $ \dfrac{\cosh(x)}{\cosh(x)} $, ...
1
vote
1answer
60 views

The convergence of $\sum_{n\geq 2}\frac{1}{n^{p}(\ln(n))^{q}}$ with two different tests.

Let $p,q\in\mathbb{R}$ and consider the series $\sum_{n\geq 2}\frac{1}{n^{p}(\ln(n))^{q}}$. i) Show by the comparison test, that the series is convergent if $p>1$ and divergent if ...
-2
votes
2answers
72 views

What methods to use to integrate $\sqrt{1+t^4}$?

I have this integral to evaluate $$\int^x_1 \sqrt{1+ t^4}\, dt$$ I have tried substitution, trig identity and integration by parts, i don't have any answer. Can anyone explain the method I need to ...
-1
votes
0answers
16 views

Can someone explain how to find the surface area of the unit sphere using Fubini's Theorem? [on hold]

Can someone explain how to find the surface area of the unit sphere using Fubini's Theorem?
2
votes
0answers
17 views

What is an intuitive/geometric definition of line integrals? Do they work in 2-dimensions?

I understand that we are finding the area of a curve given by some function f(x) over the area of another curve C. (I've also successfully plugged and chugged my way through my homework, without ...
1
vote
1answer
20 views

Solving IVP by Laplace transform

I'm trying to solve an IVP with non-constant coefficients $$ y'' + 3ty' - 6y = 1, \quad y(0) = 0, \; y'(0) = 0 $$ Taking the Laplace yields $$ s^2Y + 3(Y + sY') - 6Y = \frac{1}{s}$$ $$ Y' + ...
0
votes
3answers
37 views

How do I calculate the limit of this integral from n to n+2?

I need to find the limit, as $n\to\infty$ of $\int_n^{n+2}e^{-x^3}dx$. I tried taking the integral using integration by parts but that doesn't work so now I'm stuck.
5
votes
3answers
79 views

How can I finish integrating $\int {\sqrt{x^2-49} \over x} $ using trig substitution?

$$\int {\sqrt{x^2-49} \over x}\,dx $$ $$ x = 7\sec\theta$$ $$ dx = 7\tan\theta \sec\theta \,d\theta$$ $$\int {\sqrt{7^2\sec^2\theta - 7^2} \over 7\sec\theta}\left(7\tan\theta \sec\theta ...
1
vote
3answers
47 views

Simple Integration by Substitution requiring bizarre answer

Before addressing my queries and attemps I shall be posting the full question below. Use the substitution $x=e^u$ to find $$\int (\ln x)^2dx$$ My answer boiled down to $\dfrac{2x^3}{3} + C$ ...
1
vote
1answer
88 views

Find the antiderivative for $f(x)=\frac{1}{1+\cos^2x}$ [duplicate]

Evaluate $\int_0^x \frac{dt}{1+\cos^2t}$ $\forall x \in \mathbb{R}$ I got this question in an analysis exam, and I did what everybody does (this), I made $u=\tan t$ and I got ...
0
votes
0answers
45 views

Did I evaluate the improper integral $\int_1^2 {1 \over (1-x)^2}dx$ correctly?

$$\int_1^2 {1 \over (1-x)^2}dx$$ $$u = 1 - x$$ $$-du = dx$$ $$\lim_{a \to 0^-}-\int_0^{-1} {du \over u^2} = \lim_{a \to 0^-}\int_{-1}^{a} {du \over u^2}$$ $$\lim_{a \to 0^-} {1 \over u}|_{-1}^a $$ ...
2
votes
0answers
46 views

Differentiation Theorem

Assume that a function $f$ is integrable on $[a,b]$ w.r.t. an increasing function $g$, that $f$ is continuous at $c\in[a,b]$ and that $g$ is differentiable at $c$. Then the function defined by ...
0
votes
1answer
19 views

Consider $f(x) =3x^2+2x+a$ where a is parameter such that $\frac{da}{dt}=3$ Let $a =0$, when $t =0$ and…

Problem : Consider $f(x) =3x^2+2x+a$ where a is parameter such that $\frac{da}{dt}=3$ Let $a =0$, when $t =0$ and $A(t) =\int^t_0 \{f(x)\}\,dx$ ( where $\{\cdot\}$ denotes the fractional part ...
1
vote
0answers
22 views

Solving $\int dx {\sqrt{x^2+a}} e^{-A x^2} erf \left( c(x-b) \right)$

I got as far as: $$\int dx {\sqrt{x^2+a}} e^{-A x^2} erf \left( c(x-b) \right) $$ $$=\frac{2}{\sqrt{\pi}} \int dx \int^{c(x-b)}_0 dy {\sqrt{x^2+a}} e^{-A x^2 - y^2}$$ $$=\frac{-2 c}{\sqrt{\pi}} ...
2
votes
1answer
25 views

Transforming integral equation to differential equation

I was given the task to find all continuous functions that satisfy the following equation: $$x \int_0^x {y }dx=(x+1) \int_0^x{xy}dx$$ I am quite new to differential equations so my first thought ...
2
votes
0answers
19 views

boundary of boundary is zero (Spivak)

At the bottom of page 99 of M. Spivak's Calculus on Manifolds he arrives at the formula $$\partial (\partial c)=\sum_{i=1}^n \sum_{\alpha=0,1} \sum_{j=1}^{n-1} \sum_{\beta=0,1} ...
-1
votes
2answers
35 views

Double integrals- cartesian to polar [on hold]

$$\int^\infty_{-\infty}\int^\infty_{-\infty} \frac{1}{a^2 + x^2 +y^2}\,dy\,dx$$ How can I convert the integral to polar form the hint given in the question is:x-y plane
0
votes
1answer
35 views

Double integral (choice of) change of variables

I'm looking for a way calculate the following integral: $$\iint_D\frac{(x-y)^2(1+2y)}{(1+x+y^2} d(x,y)$$ With $D=\{(x,y)\in \mathbb{R}^2 : 0 \leq x+y^2 \leq 4 \mbox{ and } x\leq y\leq x+2\}$. what ...
0
votes
0answers
28 views

Proving if $f(x)$ is an integrable function on $[a,b]$ then $g(x)=f(x-c)$ is integrable on $[a+c,b+c]$

Prove that if $f(x)$ is an integrable function on $[a,b]$ then $g(x)=f(x-c)$ is integrable on $[a+c,b+c]$. My attempt: Since $f$ is integrable then there's a sequence of partitions ...
-2
votes
1answer
30 views

this is exercise in Rudin chapter 9 Fourier Transform [on hold]

Given $\lambda>0$ and $a\in\mathbb R$, compute $$\lim\limits_{A\to\infty}\int_{-A}^A\frac{\sin\lambda x}{x}e^{iax}dx$$