All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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9 views

Looking for advice with the following integral

I have the following integral to evaluate: $$ \int_0^t t^m (t + n)^o \sin(pt) \mathrm{d}t \quad m,n,o,p \in \mathbb{R}$$ I'm unable to proceed with this integral as it is non-trivial. Even using a ...
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1answer
15 views

Vitali Set: Inner Measure vs. Outer Measure

Context Nonlinearity in general of the Lebesgue integral for nonmeasurable functions reduces in some sense to inner and outer measure of nonmeasurable sets: ...
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0answers
18 views

Numerical value of $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

Could somebody give me a numerical value for this integral? $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$ Of course a closed-form also ...
1
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2answers
59 views

computing integral without softwares: $\int \frac{2x+3}{x^2+\sqrt{1-x^2}}dx$

I was wondering if this integral can be solve without wolfram and others: $\int \frac{2x+3}{x^2+\sqrt{1-x^2}}dx$ Thanks.
5
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2answers
60 views

Computing in closed form $\sum_{n=1}^{\infty}\frac{\operatorname{Ci}\left(\frac{3}{4}\zeta(2) \space n\right)}{n^2}$

What tools would you recommend me for computing the series below? $$\sum_{n=1}^{\infty}\frac{\operatorname{\displaystyle Ci\left(\frac{3}{4}\zeta(2) \space n\right)}}{n^2}$$ I lack the starting ...
2
votes
3answers
61 views

How to integrate $\frac{y^2-x^2}{(y^2+x^2)^2}$ with respect to $y$?

In dealing with the integration, $$\int\frac{y^2-x^2}{(y^2+x^2)^2}dy$$ I have tried to transform it to polar form, which yields $$\int\frac{\sin^2\theta-\cos^2\theta}{r^2}d(r\cos\theta)$$ But, what ...
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1answer
34 views

Integration by parts, proving inductive case

${1\over2}\int_{-\pi/2}^{\pi/2}cos^{2n-1}(x) dx$ Inductive step: Show that the $integral={(2n-2)(2n-4)...\over (2n-1)(2n-3)...}$ for $n\ge2$ $T(n+1)$=... Attempted int. by parts using ...
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0answers
59 views

Are variables the same in pure mathematics??? [on hold]

my question is In pure mathematics, $x$ always $=x$ $x = x$, the variables are abstract. In modelling, $t$ could mean the time that has elapsed since you started a machine for example. Or ...
2
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2answers
42 views

Where should I place the notorious '+c'?

Consider the following proof - $$I=\int \sin (\ln x)dx\\\iff I=\sin(\ln x)x-\int\frac{ \cos (\ln x) }{x}\cdot {x} dx \\\iff I=x\sin (\ln x)-\int\cos(\ln x)dx\\\iff I=x\sin(\ln x )-[x\cos(\ln ...
1
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3answers
36 views

finding an indefinite integral of a fraction

(a) Show that $\frac{4-3x}{(x+2)(x^2+1)}$ can be written in the form ${\frac{A}{x+2} + \frac{1-Bx}{x^2+1}}$ and find the constants $A$ and $B$. (b) Hence find ...
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2answers
42 views

Easy question on integrals

I have some problems understanding this inequality: $$\int_{x-\varepsilon x}^x \vartheta\left(t\right)dt \leq \vartheta\left(x\right)x\varepsilon$$ where $\vartheta\left(x\right)$ is the Čebyšëv (or ...
4
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1answer
63 views

Stuck on this intergral $\int^\frac{\pi}{3}_\frac{\pi}{4} \frac{\tan^2x}{x-\tan x} dx $ calculus I

$$\int^{\pi/3}_{\pi/4} \frac{\tan^2x}{x-\tan x} dx $$ this is that I have tried $$\int^{\pi/3}_{\pi/4} \frac{\frac{\sin^2x}{\cos^2 x}}{x-\frac{\sin x}{\cos x}} dx $$ $$\int^{\pi/3}_{\pi/4} ...
6
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6answers
417 views

Two methods to integrate?

Are both methods to solve this equation correct? $$\int \frac{x}{\sqrt{1 + 2x^2}} dx$$ Method One: $$u=2x^2$$ $$\frac{1}{4}\int \frac{1}{\sqrt{1^2 + \sqrt{u^2}}} du$$ ...
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1answer
21 views

Problem with simplifying before integration

Can someone explain to me how did the du = 6y^(-1/3)dy went into the last equation?
2
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0answers
9 views

Proof that maximal interval of existence exist and bounded

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
2
votes
3answers
36 views

Evaluate trig function integral

I was struggling to evaluate this integral: $$\int x\sin^2(4x)\;dx$$ Every time I try again I end up with a different answer, my most recent answer I came up with is $$-\frac1{12} x\cos^3(4x) + ...
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1answer
11 views

Given a Riemann Integrable function f, calculate the values of A,B,C [on hold]

Given a Riemann Integrable function f, calculate the values of A,B,C Any help will be thankful. Thanks!
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0answers
20 views

Change order between integral and differential calculation

Are those right? And I want to ask, in general case, when we can change the order of diff and integral: diff(integrate(L(x,y))) integrate(diff(L(x,y)))
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3answers
72 views

Evaluating the indefinite integral $\displaystyle \int 4x \sqrt{1 - x^4} dx$

I need help evaluating $$\int 4x \sqrt{1 - x^4} dx$$ What I have tried so far: Rewriting the integral as $$\int \frac{4x}{\sqrt{1 - x^4}} (1 - x^4) dx$$ $$\int \frac{4x}{\sqrt{1 - x^4}}dx - \int ...
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0answers
14 views

Integrability in bounded set

Let $A$ be a bounded open set in $\mathbb R^n$; let $f:\mathbb R^n \to \mathbb R$ be a bounded continuous function. Give an example where $\int_\bar A f$ exists but $\int_A f$ does not. Is about is ...
1
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3answers
51 views

Integration Formula

Previously, to integrate functions like $x(x^2+1)^7$ I used integration by parts. Today we were introduced to a new formula in class: $$\int f'(x)f(x)^n dx = \frac{1}{n+1} {f(x)}^{n+1} +c$$ I was ...
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0answers
8 views

Find a smooth function with prescribed moments

In several contexts I’ve encountered variants of the following problem : let $m_0,m_1,m_2$ be real numbers such that $0 < m_1 < m_0$ and $\frac{m_1^2}{m_0} <m_2 < m_0$. Then, show that ...
6
votes
5answers
122 views

How to find $\int|\cos x|\,dx$?

How do I find closed form for $\int|\cos x|\,dx$ for all real $x$? It can be expressed as incomplete elliptic integral of the second kind: $$\int|\cos x|\,dx=\int\sqrt{1-1^2\sin^2x}\,dx=E(x,1)$$ ...
1
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1answer
30 views

Integral involving exponents

How do we integrate $\int e^{C_1\frac{u^2+1}{u^2-1}} \ du\tag 1$ I could not find a proper substitution to convert it to a normal available form so that I can get a closed form of integration. $C_1$ ...
3
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1answer
89 views

Limit of the sum of $\gamma_k(x)=xf((k+1)x)-\int_{(k+1)x}^{(k+2)x}f(t)\mathrm{d}t$

Let $f$ be a continuous, decreasing function, with $\displaystyle\lim_{x\rightarrow\infty}f(x)=0$. Let $\gamma_k(x)=xf((k+1)x)-\int_{(k+1)x}^{(k+2)x}f(t)\mathrm{d}t,\displaystyle x>0$. Let ...
3
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0answers
37 views

Wicked domain of integration in a triple integral

I am dealing with a domain of integration of the form: $\left(\frac{x-y}{x+y}\right)^2+\left(\frac{y-z}{y+z}\right)^2+\left(\frac{x-z}{x+z}\right)^2\leq k$ The region looks like this (for $k=0.2$): ...
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2answers
21 views

Changing bounds on double integral

I have the following integral and with the following substitutions that I made: $$\int_{a}^{b}\int_{c}^{d}xy\sqrt{x^2+y^2}\,dx\,dy$$ $u=x^2, v=y^2$ $du = 2xdx, dv = 2ydy$ Which led me to ...
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1answer
35 views

Partial summation formula and integral

I have to prove that $\forall k \geq 1$ $$ \sum_{n\leq x} \frac{f(n)}{n} = \frac{1}{(k+1)!} \log^{k+1} x + O(\log^k x), $$ where $$ \sum_{n\leq x} f (n) = \frac{x}{k!} \log^k x + O(x\, \log^{k-1}x). ...
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0answers
13 views

A proof problem about intergral equation's root

Several days ago,my junior asked me the following problem: Let $$F\left( w \right) = \frac{1}{T}\int_0^T {M{x_C}\left( t \right)\cos \left( {tw} \right)dt} - \frac{{\sin \left( {{T_s}w} ...
6
votes
1answer
57 views

Closed-form of $\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)$

Does the following series have a closed-form \begin{equation} \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1) \end{equation} where $\Psi_3(x)$ is the polygamma function of order 3. Here is ...
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1answer
40 views

Integral of $\int\frac1{x}\sqrt[3]{\frac{1-x}{1+x}}dx$

I could substitute $t=\sqrt[3]{\frac{1-x}{1+x}}$ and get $\int\frac{6t^3}{t^6-1}dt$, which leads to partial fractions decomposition with 6 variables. That's annoying and may lead to mistakes. Is there ...
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1answer
97 views

Integral of $x^x$ [duplicate]

I can't find this integral around here, does anybody suggest how to calculate this integral? $$ I = \int x^x dx. $$ Thanks in advance
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0answers
36 views

integration involving imaginary terms

How do we integrate forms of following type with imaginary terms involved? Can we get a closed form of it as result? ...
2
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0answers
39 views

Integration using exponent

What could be the techniques we need to use to solve this integration $\displaystyle \int\tan^2\theta\frac{\sin^2(\sec\theta\tan\theta)}{\sec^2\theta}d\theta \tag1$? How do I convert this in to a ...
0
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1answer
57 views

Integration with quadratic square root

What could be the techniques we need to use to solve this integration $\int\dfrac{s^2\sin^2\left(s\sqrt{ as^2+bs+c}\right)}{as^2+bs+c}ds$ ? Main issue here is the term inside $\sin^2()$. Very ...
1
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1answer
19 views

Cumulative probability of Chi-squared distribution

If $X$ is distributed $\frac{\chi_{10}^2}{10}$ , find the probability that $X > 1.83$ The formula for the Chi-squared CDF I'm using is the following, which is the integral of the PDF formula: ...
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2answers
157 views

A Definite Integral I

Given the definite integral \begin{align} \int_{0}^{\pi} \frac{1+\cot^{2}(t)}{\cot^{2}(t)} \, \ln\left( \frac{1+2\tan^{2}(t)}{1+\tan^{2}(t)} \right) \, dt = -2\pi \end{align} then what is the general ...
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2answers
44 views

How to find indefinite integral

Im trying to find this indefinite integral: $$\int \frac{5x^2+4x-4}{(x+1)(x^2-4)}dx$$ I thought of using Partial fractions and got to $$5x^2+4x+4 = x^2(A+B+C)+x(-B+3C)-4A-2B+2C.$$ Now I have $3$ ...
4
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3answers
67 views

Integrate $\ln(2-x)dx$

I want to learn how to integrate this. If you could show me a step-by-step approach that would be awesome. If you could also point me to some good tutorials on integration that would be icing on the ...
0
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1answer
27 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 ...
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1answer
14 views

Notation regarding the continuity equation for conservation of mass

I have the following equation for the net mass flow out of a control volume through a surface $S$ - $$\int \int_S p \overrightarrow{V} \cdot \overrightarrow{d}S$$ (Actually there should be an ellipse ...
7
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0answers
53 views

Integral $\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$

I decided to follow a recent trend and ask a question about logarithmic integrals :) Is there a closed form for this integral? $$\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$$
2
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2answers
46 views

Integration of dirac function explanation

I have a problem that need your help. I have a gray image. We denotes $I(x)$ is gray level of a pixel in the image and $f(z)$ is a function of $z$(ie: histogram function...)-where $z$ is the set of ...
3
votes
1answer
54 views

Find the exact length of the curve $y=\frac 12 x^2- \frac 12 \ln(x)$

Find the exact length of the curve $y = \frac 12 x^2- \frac 12 \ln(x)$, for $2 \le x \le 4$. My attempt: \begin{align} L&= \int_2^4 \sqrt{1+\left[x-\frac 1{2x} \right]^2} \, dx \\ &= ...
0
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0answers
17 views

McShane Integral over Unbounded Intervals

I was told that the following property of the Henstock integral, $f$ is Henstock integrable on $[a,\infty]$ if and only if $$\lim_{b\to\infty}\int^b_a f$$ exists and then $$\lim_{b\to\infty}\int^b_a ...
1
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1answer
43 views

Why does this double integral give me different answers?

According to the following link: http://www.instructables.com/id/Change-of-Variables-of-Double-Integrals/?ALLSTEPS The double integral ultimately evaluates to 1.58362 after variable replacement. ...
0
votes
2answers
41 views

“Energy” of a signal

If we have a signal $$x(t)=\begin{cases} t &0\leq t < 1 \\ 0.5+0.5\cos(2 \pi t) &1 \leq t < 2 \\ 3-t &2\leq t<3 \\ 0 &\text{elsewhere} \end{cases}$$ It's energy is ...
1
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1answer
21 views

order of integrals with independent limits

I was wondering if the following is true assuming that the limits are independent (like constants) $$ \int_{\alpha}^{\beta} \int_{\gamma}^{\psi} {xy} dx dy = \int_{\gamma}^{\psi} ...
8
votes
3answers
103 views

Inequality $\int^{1}_{0}(u(x))^2\,\mathrm{d}x \leq \frac{1}{6}\int^{1}_{0} (u'(x))^2\,\mathrm{d}x+\left(\int_{0}^{1} u(x)\,\mathrm{d}x \right)^2$

I've been scratching my brain on this one for about a week now, and still don't really have a clue how to approach it. Show that for $u \in C^1[0, 1]$ the following inequality is valid: ...
1
vote
2answers
30 views

I don't understand one of the steps in solving Green's function for diffusion

Why is it that $\int_0 ^\infty u^4e^{-u^2}du = $ $ \left[\frac {d^2}{d\alpha^2}\int_0 ^\infty e^{-\alpha u^2}du \right]_{\alpha = 1} $?