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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1answer
43 views

Finding pathline

I've been trying to find the pathline of a particle dropped in a steady flow defined by the following vector components: $$ u= \frac{-2x}{(x^2+y^2+1)^2} \hat i + \frac{-2y}{(x^2+y^2+1)^2}\hat j $$ in ...
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2answers
61 views

Solutions to the integral $\int \frac {dx}{2\sqrt x (x+1)}$

I am given a question to solve the integral $\int \frac {dx}{2\sqrt x (x+1)}$. When I substitute $x+1 = t^2$, I get the solution as $\space \ln(\sqrt{x+1} + \sqrt x) +C$; while when I substitute ...
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2answers
42 views

Evaluation of an integral of some expressions involving fractions

I am stuck in evaluating the following integral: \begin{equation} \int_{0}^{b-a} \frac{1}{\sqrt{u} (a+u)} \,du, \end{equation} where $0<a<b$. Any ideas?
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1answer
59 views

Show that there is $f\in L^1(X,\mu)$ with $P(f)<\infty$ and $P(f_n-f)\to 0$ as $n\to\infty$

Could you please help me solving this old prelim problem. Any hints are appreciated
1
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1answer
98 views

An integral with density function of $N(\hat{a}, \frac{1}{s})$

I am stucked on this integral, which is from a research paper in Finance, for a while, so can anyone please help walk me through how we can get the answer on the RHS of this integral? Prove: ...
10
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1answer
260 views

Probability and Laplace/Fourier transforms to solve limits/integrals from calculus.

I've seen in some answers in Brilliant.org to some very complicated limits and integrals that uses probabilistic arguments (Let $X$ be a random variable from $[0,1]$... some examples are in those ...
5
votes
4answers
332 views

Finding $ \int_0^1 \frac {\ln x}{1+x^2}\mathrm dx $

Today I encountered the problem of how to find $$ \displaystyle\int_{0}^{1} \frac {\ln x}{1 + x^2}\mathrm dx $$ but got no start on it. Is this one of those integrals which we have to approach from ...
2
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1answer
42 views

Prove that $\int k(w)o(h^2w^2)dw=o(h^2)$ for $\int k(w)dw=1$

Suppose that $k$ is nonnegative real-valued function satisfying $$ \int k(w)dw=1,\quad\int wk(w)dw=0,\quad\int w^2k(w)dw=\kappa_2<\infty.\tag{$\star$} $$ (The limits of the integrals are all ...
8
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2answers
278 views

Closed form of $\int_{0}^{\eta}\cos nt\log\left(\frac{\cos(t/2)+\sqrt{\cos^2(t/2) -\cos^2(\eta/2)}}{\cos(\eta/2)}\right) dt$

I am reading a paper (sorry, no e-copy) with a number of infinite series, in which each term of the series is an integral of a complicated transcendental function like the one in the title. There ...
4
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3answers
125 views

Find $ \int \frac {1-x^2}{1+3x^2+x^4} \, \mathrm{d}x $

Today, the CalcBee sample problems got released. The following problem was my creation and I wanted to see how many solutions people can come up with. The result is very beautiful and I thought it ...
5
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4answers
182 views

Evaluating $\int{\frac{1}{\sqrt{x^2-1}(x^2+1)}dx}$

Evaluating $$\int{\frac{1}{\sqrt{x^2-1}(x^2+1)}dx}$$ using $ux=\sqrt{x^2-1}$ UPDATE 'official' solution $$u^2x^2=x^2-1$$ $$x^2=\frac{-1}{u^2-1}$$ $$x^2+1=\frac{u^2-2}{u^2-1}$$ ...
8
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2answers
229 views

Closed form of $\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln x}{(1-x)^3}\,\mathrm dx\;\mathrm dy\;\mathrm dz$

While trying to find several references to answer Pranav's problem, I encounter the following multiple integrals $$I=\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln ...
8
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2answers
134 views

How find this integral $I=\int_{-1}^{1}\frac{dx}{\sqrt{a^2+1-2ax}\sqrt{b^2+1-2bx}}$

show this integral $$I=\int_{-1}^{1}\dfrac{dx}{\sqrt{a^2+1-2ax}\sqrt{b^2+1-2bx}}=\dfrac{1}{\sqrt{ab}}\ln{\dfrac{1+\sqrt{ab}}{1-\sqrt{ab}}}$$ where $0<a,b<1$ my idea: let ...
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1answer
74 views

Check my answer - simple laplace transform of piecewise continuous function.

I'd just like to check that I got the idea right, first exercise im doing in laplace transforms and am a bit clueless. We are given $f(t)=0$ if $0<t<2$ and $f(t)=t$ if $t>2$. We are asked to ...
0
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1answer
61 views

If a sequence $(f_n)$ converges in $L^2$, then $g'(x)\int_0^x f_n(t)\,dt$ converges in $L^1$

The first: Suppose $g$ is increasing and differentiable on $[0,1]$. For every $f\in L^2(0,1)$ define $f^*(x)$, for $x\in [0,1]$, by: $$f^*(x)=g'(x)\int_0^x f(t)\,dt .$$ If $f_n\to f$ in $L^2(0,1)$, ...
2
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1answer
41 views

Why $ \int_0^{\infty} du \, \frac{e^{-3 u} - e^{-4 u}}{u} = \int_0^{\infty} du \, \int_3^4 dt \, e^{-u t} \\ $?

from this answer I could not see what is happening here: $$ \int_0^{\infty} du \, \frac{e^{-3 u} - e^{-4 u}}{u} = \int_0^{\infty} du \, \int_3^4 dt \, e^{-u t} \\ $$ What technique of integration ...
2
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1answer
63 views

How find this sum closed form $I=\sum_{k=1}^{n}\int_{0}^{+\infty}\cos{(2kx)}x^{m-1}e^{-ax}dx$

Find this closed form? $$I=\sum_{k=1}^{n}\int_{0}^{+\infty}\cos{(2kx)}x^{m-1}e^{-ax}dx,m\ge 1,a>0$$ use ...
4
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0answers
72 views

Seperating single integral into an double integral.

Please refer to : How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$ The answer by @Venus. What is the procedure in converting that ...
9
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1answer
155 views

Multiple integrals involving product of gamma functions

The following integral was posted a few days back on Integrals and Series forum: $$\int_0^{2\pi} \int_0^{2\pi} \int_0^{2\pi} \frac{dk_1\,dk_2\,dk_3}{1-\frac{1}{3}\left(\cos k_1+\cos k_2+ \cos ...
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3answers
90 views

Seemingly Simple Integration: $x/(x-1)$ [closed]

I am currently working on some advanced engineering math but this seemingly simple integral has me stuck. Someone please show me how to derive it. It is part of a far bigger more complex problem in ...
2
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2answers
29 views

Find the continuous function such that the Riemann integrable is the same

Find all functions $f$ such that $f$ is continuous on $[0,1]$ and $\int_0^x f(t) dt = \int_x^1 f(t) dt$ for every x $\in (0,1)$ I can't think of any function that would satisfy this property! ...
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1answer
49 views

What is the 'largest' space of integrable functions which is also a Hilbert space?

It is well known that $L^2(X,\mu)$, the set of functions $f:X \rightarrow \mathbb{C}$ such that $\int_X |f|^2 \text{d} \mu < \infty$, is a Hilbert space. Is there a Hilbert space $H$ such that ...
6
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1answer
237 views

How evaluate the following hard integrals?

Prove: $$\displaystyle\int_0^{\frac{\pi}{4}}{\,x}{\,\arctan\sqrt{\frac{\cos2x}{2\sin^2x}}}dx=\frac{\pi}{96}[{\pi^2}-6\ln^22]$$ And ...
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2answers
113 views

Surface area of a solid of revolution: Why does not $ \int_{b}^{a} 2\pi \,f(x) \,dx $ work?

Why does not $ \int_{b}^{a} 2\pi \,f(x) \,dx $ yield the correct answer when calculating the surface area of a solid of revolution?
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0answers
40 views

Clarifying a step in an integration solution

In the accepted answer here, the first two steps in computing the integral are \begin{align} \mathcal{I} =&\frac{1}{2}\int^\infty_0\ln(1-e^{-2x})\ln\left(\frac{x^2}{\pi^2+x^2}\right)\ {\rm d}x\\ ...
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1answer
57 views

Integration problem that may use DCT

I am trying to solve the following problem. Let $f \in L^2(0,1)$ and define $$f_n(x)= n \int\limits_{k/n}^{(k+1)/n} f(y) dy $$ for $x \in [k/n, (k+1)/n)$, $k=0,1, \dots, n-1.$ Show that $f_n ...
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2answers
42 views

If $f_n$ converges to $f $ in $p$-norm, then $f_n$ converges to $f$ in measure.

I want to prove that if $f_n$ converges to $f $ in $p$-norm, then $f_n$ converges to $f$ in measure. This is the proof: Suppose not. Then there exist $\epsilon>0,\delta> 0$ such that $μ \{x: ...
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3answers
57 views

Fundamental Theorem of Calculus 1 - definite integral

I have two problems, they're not from a book so I can't check the answer for one of them and the other I'm not sure on what to do. $$ {d\over dx}{\int^{1}_{x^{2}}} {\sqrt{t^{2}+1}} {dt} $$ $$=-{d\over ...
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4answers
82 views

Expectation of non-negative random variable

Let $X$ be a non-negative random variable. In a proof for $E[X]=\int_0^\infty P(X>t)dt$ from the answer of this question, we use Fubini for the middle quality. Why do we need $X$ to be ...
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2answers
34 views

Bound on and integral

If $\alpha \in \Bbb R$, how can I show $$\int_{-M}^M \frac{1}{\sqrt{|x-\alpha|}} \, dx \le 4 \sqrt{M}$$ For $M>0$, Rewriting the integral gives $$\int_{-M+\alpha}^{M + \alpha} ...
1
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0answers
49 views

product of barycentric coordinates over a simplex

Show that: $$\int_K \eta_0^{\alpha_0}(x)\cdots\eta_n(x)^{\alpha_n}dx=\frac{{\alpha_0}!\cdots{\alpha_n}!n!|K|}{(\alpha_0+\cdots+\alpha_n+n)!}$$ where $\eta_i$ barycentric coordinates and $K$ is a ...
2
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3answers
34 views

Determine monotone intervals of a function

Let $$ f(x) = \int_1^{x^2} (x^2 - t) e^{-t^2}dt. $$ We need to determine monotone intervals of $f(x)$. I tried to differentiate $f(x)$ as follows. $$ f'(x) = \left(x^2 \int_1^{x^2} e^{-t^2}dt \right)' ...
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0answers
37 views

Finite integral involving branch cut. Basic Argument Question

I am reading this Wikipedia article on examples of contour integrals using complex analysis (http://en.wikipedia.org/wiki/Methods_of_contour_integration). In particular, I am looking at Example (VI), ...
2
votes
8answers
140 views

How to show that $f(x) = 0$ if $\int_a^bf(x)\,\text{d}x=0$ for all $a,b\in\mathbb{R}$?

I found this problem on the web: Let $f(x)$ be a real-valued, continuous function with the property that $$\int_a^bf(x)\,\text{d}x=0$$for all real numbers $a,b$. Prove that $f$ is identically $0$. ...
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6answers
165 views

Proving that $\int_0 ^1 \frac{\text{d}s}{\sqrt{1-s^2}}$ converges with no trig functions

Let $$\int_0 ^1 \frac{\text{d}s}{\sqrt{1-s^2}}$$ How to show that it converges with no use of trigonometric functions? (trivially, it is the anti-derivative of $\sin ^{-1}$ and therfore can be ...
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0answers
24 views

How to calculate convolution of function defining a measure

Given the function $F(t)=2-2e^{-t}$ defining a measure on $(\mathbb{R}_+,\mathfrak{B}(\mathbb{R}_+))$ and I want to calculate the convolution of this function with itself. I tried to do that by using ...
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1answer
52 views

How to compute $\int^{1}_{-1}f(x)dx$?

I need to compute $\displaystyle\int^{1}_{-1}\,{\rm f}\left(\, x\,\right)\,{\rm d}x$, where $$ \,{\rm f}\left(\, x\,\right) =\left\{\begin{array}{lcrcl} x & \mbox{if} & x & \leq & 0 ...
19
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3answers
609 views

Evaluate $\int_0^{\pi/2} \frac{\ln\left(e^{2x} + 1\right)}{1 + \sin2x}\mathrm dx$

Here's my Xmas gift to all of you! I just encountered a very tough integral. $$\int_{0}^{\pi/2} \frac{\ln\left(e^{2x} + 1\right)}{1 + \sin2x}\mathrm dx$$ I have tried for a few hours. This task is ...
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3answers
163 views

How to solve the differential equation $(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$?

Solve $$(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$$ I tried the substitution $y^2=t$ ; $2y\:\text{dy}=\text{dt}$ to get $$(x^3)\:\text{dt}+(1-t)[(x^2+1)t-1]\:\text{dx}=0$$ ...
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2answers
121 views

How to integrate a fraction of the type $\frac{1}{(ax+b)^c(dx+e)^f}$?

I'm working on obtaining chemical reactions' speed, and this is one of the problems I met with. $$ \int \frac{1}{(ax+b)^c(dx+e)^f}dx $$ Can this equation could be solved? If possible, please show ...
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1answer
33 views

Is this enough to demonstrate divergence of an improper integral?

The integral in question is $$\int_0^\infty (f(x)-a)^2dx$$ Where f(x) is some continuous function and a is some constant. When we expand the integrand,we end up with an $a^2$ term. We can then ...
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3answers
114 views

Evaluation of the integral $\int 3x \cos x^2 \, dx$

I want to solve this: $$\int 3x \cos x^2 \, dx$$ I get this answer: $$ \frac{\sin 2x}{2}+\frac{\cos 2x}{4}+C $$ but the answer should be: $$ \frac{3 \sin x^2}{2}+C $$ Am I doing anything wrong ...
0
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1answer
73 views

How to approach, substitution - definite integral

So I have this problem $${\int^{\pi/2}_0} {{\cos\theta \sin\theta}\over \sqrt{\cos^{2}\theta +8}}d\theta $$ and I'm not sure if this is the right direction to begin. If I have $u = \cos\theta$ ...
1
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1answer
118 views

A question from my final exam

Today I had the final exam of the lesson Mathematics I. There was a question that I want to know if I solved correctly the following limit. $$\lim\limits_{n\to\infty}\left(\frac{n}{0^2+n^2} + ...
2
votes
1answer
87 views

Evaluating $\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$

How to calculate this integral? $$\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$$ I suppose that it should be parted like this: $$\int_0^{1} \frac{dx}{\sqrt[3]{2x^2-x^3}} + \int_1^{2} ...
1
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2answers
61 views

Generating functions for $\log^3(1-x)$ of $\log^3(x)$

I am trying to find generating functions which will give me a power logarithm. I am trying to find generating sums in the form $$\sum_{n=1}^{\infty} a_n\,x^n = -\frac{\log^2(1-x)}{1-x}$$ or ...
1
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2answers
57 views

Taylor expansion of the Error function

The error function $\operatorname{erf}(z)$ is defined by the integral $$ \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2}\,dt,\quad t\in\mathbb R$$ Find the Taylor expansion of ...
2
votes
1answer
86 views

Composition of a Dirac delta and a function in higher dimensions

Coming from a physics background, I was taught the formula for the composition of a Dirac delta and a function. Indeed, if we consider a nice function $ f : \mathbb{R} \to \mathbb{R} $, one can write ...
2
votes
1answer
66 views

Integral over a simplex

Let $C_k$ be the $k$-simplex. I know that $$\int_{C_k} \prod_{i=1}^k x_i^{\alpha_i-1} dx_i = \frac{\prod_{i=1}^k \Gamma(\alpha_i)}{\Gamma\left(\sum_{i=1}^k \alpha_i\right)} \equiv ...
4
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2answers
61 views

Calculating Triple Integral

I have task : find volume of body limited by surface $(\frac{x}{a})^{2/3} + (\frac{y}{b})^{2/3} + (\frac{z}{c})^{2/3}$ = 1. I know that this task is about triple integral. But i have confused by such ...