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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1answer
34 views

Integral equation, Fourier transform

Find all functions $ f : \mathbb{R} \rightarrow\mathbb{R} $, that solve $\int_{-\infty}^{\infty} f(t-x)f(x) dx =e^{-t^2}$, $ t\in \mathbb{R}$ How do I solve this? I know that the left part is the ...
0
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0answers
28 views

Line Integral over triangle/force field

having some trouble with some line integrals: Compute the line integral of $F = <e^z, e^{x-y}, e^y>$ over the path from $A(2,0,0)$ to $B(0,4,0)$ to $C(0,0,6)$ to $A$. Thanks!
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1answer
27 views

What are the consequences of this simple property of $L^1$ functions?

I came across the following statement: Let $f\in L^1(\mathbb R,\mathbb R)$. Then $$\forall \varepsilon>0 \ \ \exists \delta>0 \ \ \text{such that for all open sets } U\subset\mathbb R \text{ ...
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1answer
26 views

Finding volume of convex polyhedron

I am trying to compute the volume of the convex polyhedron with vertices (0,0,0), (1,0,0),(0,2,0),(0,0,3) and (10,10,10). I am supposed to use a triple integral but am struggling with how to set it ...
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2answers
87 views

Why this integral is confusing?

I am a bit confusing. What is the value of this integral? $$I(x)=\int0\mathrm{d}x$$ Is it $0$ or $C$ where $C$ is a constant? We have: $\mathrm{d}I(x)/\mathrm{d}x=\mathrm{d}C/\mathrm{d}x=0$
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1answer
39 views

Are the two integrals equivalent?

Consider $x \in \mathbb{R}$, $A\subseteq \mathbb{R}$, $f(x)$ continuous in $\mathbb{R}$ and the integral $$ g(A):=\int_{x \in A}^{} f(x) dx $$ Is $g(A)$ equal to the integral $$ ...
2
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2answers
149 views

What is the easiest way to integrate $y=\frac {x+4}{\sqrt{-x^2-2x+3}}$?

What is the easiest way to integrate $y=\frac{x+4}{\sqrt{-x^2-2x+3}}$ ? I tried to integrate it by making numerator in form: $-2x-2$ and then pulling it under differential, but the result drastically ...
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4answers
261 views

Does the improper integral exist?

I need to find a continuous and bounded function $\mathrm{f}(x)$ such that the limit $$ \lim_{T\to\infty} \frac{1}{T}\, \int_0^T \mathrm{f}(x)~\mathrm{d}x$$ doesn't exist. I thought about ...
3
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1answer
21 views

Convergence in average on every set implies convergence?

Let's say we're working in a measure space $(X, \mathcal{B}, \mu)$, and let $f_n, f$ be measurable. Suppose I have that, for any measurable set $E$, $$ \int_E f_n d \mu \to \int_E f d \mu $$ Does that ...
2
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2answers
122 views

Cool Integral $\int_0^{\pi/2}dx\ln \sinh x$

$$ I_1=\int_0^{\pi/2}dx\ln \sinh x,\quad I_2=\int_0^{\pi/2}dx\ln \cosh x, \quad I_1\neq I_2. $$ I am trying to calculate these integrals. We know the similar looking integrals $$ \int_0^{\pi/2}dx\ln ...
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1answer
89 views

Integrate $\int^{ln(2)}_0 (3e^u - e^{2u} - 2)\sin(nu)du$

I'm having trouble integrating this function $$\begin{equation} \begin{split} f(x) & = \int^1_0x(1-x)\sqrt{1+x}\sqrt{1+x}\sin(n \ln(1+x))/[(1+x)^2] = \\ & = ...
1
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1answer
35 views

positive measurable function on $[0,1]$

If $f$ is a positive measurable function on $[0,1]$, which is larger, $$\int_{0}^{1}f(x)\,\log f(x)\,dx \qquad \text{or} \qquad \int_{0}^{1}f(s)\,ds\int_{0}^{1}\log f(t)\,dt$$ Can you help me ...
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1answer
30 views

Show the improper integral $\int^{\infty}_0 \frac 1 y e^{-y} dy$ doesn't converge.

Show the improper integral $\int^{\infty}_0 \frac 1 y e^{-y} dy$ doesn't converge. Using Wolfram Alpha: http://www.wolframalpha.com/widgets/view.jsp?id=8ab70731b1553f17c11a3bbc87e0b605 the result ...
2
votes
2answers
46 views

Integral along $\Gamma_c := \{c + i t \mid c>0 , -\infty < t < \infty\}$

I have a Complex Analysis homework problem which I've been working on for some time, and have become stuck. I am asked to compute $$ I \equiv {1 \over 2\pi{\rm i}}\int _{\Gamma_c}{a^{s} \over ...
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1answer
50 views

Help with finding the definite integral of $e^{\frac{2x-x^2}{2}}$?

I have this integral that I am trying to evaluate by hand, but I am encountering some difficulties. According to Wolfram Alpha, the answer seems to be: However, I do not understand how they got ...
2
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0answers
56 views

Integral $\int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx$

Hey I am trying to integrate $$ \int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx,\quad \alpha,n \geq 1. $$ Thanks. This integral is old and has a 4th order pole. I am also looking for literature ...
4
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2answers
75 views

Limit of the integral of $\frac{x^n+1}{x^n+2}$

Consider the following integral: \begin{align} F(x) = \lim_{n\rightarrow\infty}\int f_n(x)dx = \lim_{n\rightarrow\infty}\int\frac{x^n+1}{x^n+2}dx \end{align} How does one evaluate this integral ...
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2answers
65 views

Please, help with this integral.

$$\int \frac{\sqrt{x^2-1}}{x^5\sqrt{9x^2-1}}\; dx $$ Thanks for your help.
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1answer
42 views

Prove $\ln\big(\frac{21}{10}\big) \leq \sum_{n=10}^{20} \frac{1}{10n}\leq \ln\big(\frac{20}{9}\big).$

Please help. I have been trying his question for the past 2 hours and cant seem to go anywhere with it. $$ \ln\big(\frac{21}{10}\big) \leq \sum_{n=10}^{20} \frac{1}{n}\leq \ln\big(\frac{20}{9}\big). ...
7
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2answers
165 views

Fun Integral $ \int \frac{dx}{\cos^3 x+2\sin(2x)-5\cos x}$

$$ I\equiv \int \frac{dx}{\cos^3 x+2\sin(2x)-5\cos x}. $$ This integral does have a closed form. I am not sure where to start. We can factorize the denominator as $$ \cos^3 x+2\sin(2x)-5\cos ...
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0answers
28 views

Integral$\int_{-\infty}^\infty x^{2n} e^{-\beta (x^2+\cos x+\alpha x)}dx$

Hi I am trying to integrate $$ \int_{-\infty}^\infty\int_{-\infty}^\infty (xy)^{2n}\exp\left({-\beta(x^2+y^2+\cos x+\alpha x+iy)}\right)dxdy \quad \alpha,\beta,n >0. $$ These integrals can be ...
1
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1answer
35 views

Integral $\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n$

$$ I\equiv\mathcal{F}_n(z)=\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n. $$ Evaluate I for $n \to \infty$ and z real. We can consider $z\geq 0$ due to the symmetry of $\mathcal{F}$ given by $$ ...
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0answers
44 views

Integrate $ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx$

Hey I am trying to integrate $$ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx,\quad \alpha,n \in \mathbb{R}^{0+}. $$ This integral is old and has a triple pole. I am also looking for ...
0
votes
1answer
12 views

Work done by a vector field along a curve

I have to find the work done by a vector field $\vec{F}$ along a curve $C$ from $t = 0$ to $t = 1$: $$ \vec{F}(x,y,z) = (2xe^y + 2x)\vec{i} + (x^2e^y)\vec{j} + (3z^2)\vec{k} \\ C: \;\vec{r}(t) = ...
3
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2answers
80 views

Why this equation is true?

Pardon my ignorance. I don't know enough calculus to understand this. I assume this is a very easy question for this amazing site. I saw this on the The Theory of Riemann Zeta Function Book. ...
0
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0answers
39 views

Why is $\frac {dy}{dx}$ treated as a fraction? Plus an implicit differentiation question. [duplicate]

Why is $\frac {dy}{dx}$ treated as a fraction? I always thought that it is just notation for the derivative of $y$ with respect to $x$, but when it comes to implicit differentiation and integration ...
0
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3answers
129 views

Compute the Integral

Compute the integral. $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$ The answer at the back of the book is $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$
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5answers
422 views

Solution of a integral

$$ \int e^x \, \left(1 + \frac{e^{-x}}{x} \right) \,dx $$ I got three different integrals from this one, which are integral of $e^x$, integral of $1/x$ and the third one is integral of $e^{-x}/x$ but ...
0
votes
4answers
37 views

Integrate the given equation

Here is the equation that has to be integrated plus my answer: $$ \int \frac{x}{2x^2-1}dx=\frac12\cdot\frac23\ln{|1-2x|} +C $$ The correct answer: $$\frac14 \ln{|1-2x|} + C$$ How it come? Thanks
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1answer
35 views

Integrate the following

Here is the equation that has to be integrated and my ans: $$\int x\left(x^2+1\right)^3dx=\int \left(x^3+x\right)^3=\left(x^3+x\right)^4\dfrac 1{12}+c$$ The correct answer is : $$ \int x(x^2+1)^3 ...
2
votes
2answers
52 views

Why are differential of $\sin^2(x)$ and integral of $\sin(2x)$ not the same?

I was working on a list of common integrals and differentials and I came across this question. If $${d\over d\theta}(\sin^2\theta) = \sin(2\theta)$$ Then why is $$\int \sin(2\theta) \space d\theta = ...
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3answers
41 views

Changing limits of integration

Is that true, that for every function $f(x)$ changing the limits for opposite, causes: $$\int^0_{-a} f(x) dx = \int_0^a f(-x) dx$$
2
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0answers
41 views

Integral of Difference of Logs

I get the expansion of $h$ to be $$ h(z) = {1 \over z } \sum_{r=1}^{\infty}{1 \over r}{(-{\alpha \over z}})^r $$ $$ \Rightarrow h(z) = \sum_{r=-2}^{-\infty}{{(-\alpha)^{r+1} \over -(r+1)} z^{r}} $$ ...
0
votes
1answer
24 views

Example of a Riemann integrable function which is not a simple function

I'm looking for an example of a Riemann integrable function which isn't simple? I know that all simple functions $f: I \rightarrow E $ ( where $I \subset \mathbb{R}$ is an interval and $E$ is a ...
0
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0answers
25 views

How does this integration by parts work: $\int_{Q}v\varphi_t\;dxdt = -\int_S \varphi v|_{S} \nu_t - \int_Q v_t \varphi\;dxdt$

Let $\Omega(t)$ be a bounded domain for each $t$. Let $Q=\bigcup_{t \in [0,T]} \Omega(t) \times \{t\}$ and $S=\bigcup_{t \in [0,T]} \partial\Omega(t) \times \{t\}$. The normal vector to $S$ at ...
0
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1answer
55 views

Integrate square root of 4th grad polynomials

During some calculations for a program I came upon this Integral which I am not able to solve. I already tried Matlab but it didn't help me. Here is the Integral: $$\int\left(\sqrt{\sum_{0}^{5} 9 ...
1
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1answer
36 views

Finding the mean with absolute value

This question is out of my field and topic that I am teaching myself now, but I was wondering how would you solve this problem if it had the absolute value of it. My Question: $$f(x) = ...
1
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2answers
37 views

Learning integration, and have gotten to a certain point. What is the next logical step?

I've learnt the following in terms of integration; What are integrals, and what do they represent? Indefinite integrals as the opposite of derivatives Using the power rule for derivatives to provide ...
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1answer
40 views

Verify the given function including the integral $e^{-x^2}$

I'm really stuck trying to verify that the given function is a solution of the differential equation. I've attempted applying converting it to polar coordinates but I don't think I'm on the right ...
0
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1answer
34 views

Integral $ \int_{-\infty}^{\infty} e^{ \frac{-u^2}{2}} du$ [duplicate]

Could you explain what rules are used to compute this integral? $ \int_{-\infty}^{\infty} e^{ \frac{-u^2}{2}} du = \sqrt{2\pi}$
2
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3answers
137 views

Integral of 1/(8+2x^2)

I have been following a rule saying that $$\int{\frac{1}{a^2+u^2}}dx = \frac{1}{a}\tan^{-1}(\frac{u}{a})+c$$ The question is asking for the interval of $$\frac{1}{8+2x^2}$$ Following that rule ...
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2answers
55 views

Integral of $\frac{-2}{\sqrt{16-x^2}}$

So I am asked to find the anti-derivative of $$\frac{-2}{\sqrt{16-x^2}}$$ First step I took was to make it easier for me to visualise $$\int{-2(16-x^2)^{-\frac{1}{2}}}dx$$ I let $u = 16-x^2$ So ...
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1answer
16 views

Linear algebra with integrations and primitive functions. What is the area?

The line $y = kx + m$ tangents the line $y= \frac12x^3$ where $x = -2$. Decide the area of the area that is limited by the lines and the curve with the help of integration. (or any other way with ...
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0answers
16 views

Variation on Fubini's Theorem

My attempt: Let $P_1$ be a regular partition of $R_1$ and $P_2$ a regular partition of $R_2$. Denote by $P$ the corresponding regular partition of $R_1\times R_2$. Given a generalized rectangle ...
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0answers
32 views

Derivative of an integral where the variable of derivation is in the integral

How does one deal with finding $$\frac{d}{dx} \int_{0}^{x} x \, dt$$ or any integrand function that is a constant relative the integration but a variable in the derivation? Thanks.
2
votes
1answer
56 views

Prove $\sum^{\infty}_{n=1} \frac{a_{n}-a_{n-1}}{a_{n}}=\infty$

Prove $$\sum^{\infty}_{n=1} \frac{a_{n}-a_{n-1}}{a_{n}}=\infty$$ Where $a_{n}$ is an increasing sequence of positive terms that goes to infinity. I tried to approach it with $\log(a_{n})$ like a ...
3
votes
1answer
120 views

Integrate $ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx $

Hi I am trying to come up with a closed form expression for $$ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx,\quad p\geq 0. $$ I am interested in this general case in terms of p. For small p, we can ...
1
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4answers
57 views

Integrating a function with substitution

Totally forgot how to integrate. $$ \int \frac{1}{x^2 \sqrt{x^2+4}}dx$$ Just need a tip, for this what would I use to substitute?
2
votes
1answer
22 views

Is there a set formula for integration like there is for derivatives?

I know that the derivative of $f(x)$ must be $$f'(x)=\lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ We can use this formula to derive the derivatives of some functions like $\sin(x)$. Is there such a ...
1
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0answers
17 views

Definition of a Regular Partition of a Closed Generalized Rectangle in $\mathbb R^n$

What the heck does this definition of a regular partition $P$ of $R$ mean? I follow what it is saying until we get to the last part, "the $k_1\cdot k_2\cdot \cdots \cdot k_n$ subrectangles of the ...