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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33 views

Theorem to show trajectories of differential equations are close after small change to initial condition

Consider two solutions(or trajectories), say $x_1(t)$ and $x_2(t)$, of a system of differential equaions. That is, $$ x_1'(t)=x_2'(t)=f(x,t), t\ge0. $$ Also, $\|x_2(0)-x_1(0)\|<\epsilon$ for some ...
4
votes
4answers
253 views

Can we determinine the convergence of $\int_0^\infty \frac{x^{2n - 1}}{(x^2 + 1)^{n + 3}}\,dx$ without evaluating it?

Can we determine convergence without evaluating this improper integral? $$\int_0^\infty {\frac{x^{2n - 1}}{{\left( x^2 + 1 \right)}^{n + 3}}\,dx}\quad\quad n\geq 1\;,\; n\in\mathbb{Z}$$ When ...
0
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1answer
22 views

determine consumer and producer surpleus

Demand equation: $$p=60-\frac{50q}{\sqrt{q^2+3600}}$$ Supply equation: $$p=10\ln(q+20)-26$$ Determine consumers’ surplus and producers’ surplus under market equilibrium. Round your answer to the ...
0
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1answer
36 views

How to compute $\int_{-\infty}^{\infty} | e^{-(y-a)^2/2}-e^{-(y+a)^2/2}| dy$

I am looking on how to compute or a table of integral that has solution to \begin{align*} \int_{-\infty}^{\infty} | e^{-(y-a)^2/2}-e^{-(y+a)^2/2}| dy \end{align*} Using Wolfram-alpha I found it to be ...
4
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3answers
112 views

Proof of this definite integral?

Saw this sometime in my calculus book, from the Putnam Math Challenges listed: $$\lim _{ n\rightarrow \infty }{ \int _{ 0 }^{ 1 }{ \int _{ 0 }^{ 1 }{ \underbrace{\dots}_{n-3 \, times} \int _{ 0 }^{ ...
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1answer
24 views

Let X, Y be independent random variables and each one has E(1)(exponential distribution). Prove that $W_1, W_2$ are independent.

$W_1=\min \{X,Y\}$;$\ \ W_2=X-Y$ It's given that the density functions for $(X,Y),W_1$ and $W_2$ respectively are: $$f_{(X,Y)}(x,y)=e^{-(x+y)};f_{W_1}(u)=2e^{-2u}, u>0,f_{W_2}(v)={e^{|v|}\over ...
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0answers
30 views

Solving integrals with delta function constraints

What is the best way to solve integrals which use delta function constraints/restrictions? For example if I have the integral $\int_V ...
1
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1answer
53 views

Tough definite integral

Does anyone know the exact value of this integral? $$\displaystyle \int^1_0 \dfrac{p^2-1}{\text{ln } p}dp$$ I know that the approximate value is $1.09861...$. But I can't seem to get a figure on ...
1
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1answer
20 views

Probability denisity function of simple harmonic

Suppose that a spring is oscillating up and down with vertical position given by $u(t) = \sin(t)$. If you pick a large number of random $t$ to look at the position, then prove that the PDF is ...
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1answer
24 views

Estimation of the integral

I am trying to compute, or find a good estimate from above the following integral $$ \frac{1}{\pi}\int_{-\infty}^{\infty}|t|^{-1/p}\left|\frac{|t|^{\nu}-1}{t-1}\right|dt, $$where $0<1/p<1$ and ...
2
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2answers
36 views

$\int_2^x\frac{dt}{\log^kt}=O\left(\frac{x}{\log^kx}\right)$

I seek to prove the identity $$\int_2^x\frac{dt}{\log^kt}=O\left(\frac{x}{\log^kx}\right)$$ I was given the following hint: Split the integral into $\int_2^{f(x)}+\int_{f(x)}^x$ for a ...
2
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2answers
68 views

Manipulating $\int_{x_{0}}^{\infty} \frac{1}{x} \, \cos (x t) \, \text{e}^{-x^{2}} \, dx$

Is there a way to express the integral $I(x_{0}, t) = \int_{x_{0}}^{\infty} \frac{1}{x} \, \cos (x t) \, \text{e}^{-x^{2}} \, dx$, where $x_{0} \neq 0$ and $t \ge 0$, in terms of more well-known ...
1
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0answers
40 views

How to do integration by parts with brownian motion?

I am not sure how to perform integration by parts in the following expression: $$ \left(1-t\right)\left(B_t - B_s + \int_s^t \frac{r}{1-r} \mathrm{d} B_r \right) $$ Can anyone help me to solve this ...
0
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1answer
37 views

Find the distribution of $ Z=|X-Y|$ if $X=U(0,1), Y= U(0,3)$ X,Y- independent random variables.

Find the distribution of $ Z=|X-Y|$ if $X=U(0,1), Y= U(0,3)$ X,Y- independent random variables... K-relevant are. $W_1$-area bounded by $y_1, y_2$. $W_2$-area bounded by $y_3$ from above, $W_4$- area ...
2
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1answer
32 views

Integrate a partial derivative

If we define the operator $\mathcal{G}f(t,x)= \frac{\partial f}{\partial t}(t,x)$, what is the value of $$ \int_0^t \mathcal{G}f(s, b(s)) ds? $$ I'm sure it's some subtlety in the Fundamental Theorem ...
1
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0answers
27 views

Integrating a product of to error functions and an exponential

I have the following integral that I need to solve. $\int_{-\infty}^\infty \exp(-\frac{x^2}{2})*\text{erf}(x-\delta)*\text{erf}(x-\gamma)dx$ I was hoping I could use this: Integral of product of ...
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0answers
11 views

Proving that the Jacobian for spherical coordinates and the Jacobian for integrating over a sphere coincide

I want to use the formula for switching an integration from spherical to cartesian coordinates : $\int_{RS^{n-1}} \! f(x) \, \mathrm{d}\sigma (x)$=$\int\int...\int ...
3
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2answers
30 views

Limit of sequence and Riemann sum

I have to calculate $$\lim_{n \to \infty}\sum_{k=1}^{n}\frac{(k-1)^7}{n^8}$$ So, $$\lim_{n \to \infty}\sum_{k=1}^{n}\frac{(k-1)^7}{n^8} = \lim_{n \to ...
2
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0answers
831 views

Solving an integral for a characteristic function

For $L>0, H>0,\alpha>0,\sigma>0,$ $$f(t)=\int_L^H \frac{ e^{i t x} \alpha H \left(\frac{\sigma -H \log \left(\frac{H-x}{H-L}\right)}{\sigma }\right)^{-\alpha -1}}{\sigma (H-x)} \, ...
0
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1answer
23 views

Integral transformation

I'm trying to do a transformation of an integration, I have that $$\int_{0.5}^1\int_0^{0.5}e^{xy}xydxdy$$ And I want to get that integrate $$\int_0^1\int_0^1 f(x,y)dxdy$$ Where the value of the ...
1
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1answer
29 views

What do you call a frequency that varies by a function?

I have a concept that I need to learn more about, but I don't know what it's called so I'm not sure what search terms to use to look for it. I apologize in advance that while I'm comfortable with ...
1
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0answers
43 views

Integration on an exponential function

I am struggling to solve this expression. I want to show that, $$\frac{1}{p}\nabla_{j}\int e^{ipR\cos(\theta)} dT=i\int \hat{p_{j}} e^{ipR\cos(\theta)} dT$$ here, $dT=d(\cos(\theta))d\phi$ I tried ...
3
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1answer
96 views

Integral Inequality Proof Using Hölder's inequality

I'm working on the extra credit for my Calculus 1 class and the last problem is a proof. We have done proofs before, but I'm unsure of how to approach this problem. Any help would be much appreciated, ...
2
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6answers
318 views

How to integrate when integration by parts never ends?

So I have the following integration: $\int{e^{-x}\sin{x}}$ I let u be the $e^{-x}$ and $dv = \sin{x} dx$ After doing the integration by parts I get:$(e^{-x})(-\cos{x})-\int{-e^{-x}*(-\cos{x})}$ ...
1
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2answers
34 views

Videos for finding the area under curve using integrals?

Can anybody point me in the direction of some good videos for finding areas under curves using integrals? Currently studying for a calc 1 final, have found good videos on khan academy and ...
1
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1answer
37 views

Limit of sequence and Riemann sum problem work verification

I have to calculate $$\lim_{n \to \infty}{\sum_{k=1}^{n}{\frac{n}{k^2-4n^2}}}$$ My attempt: $$\lim_{n \to \infty}{\sum_{k=1}^{n}{\frac{n}{k^2-4n^2}}} = \lim_{n \to ...
1
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1answer
47 views

For $(n-1)$-form $\omega$ on $M^{n}$ compact, orientable without boundary, then $d\omega$ vanish for some point

Let $M^{n}$ manifold compact, orientable without boundary and $\omega$ $(n-1)$-form then there is $p\in M$ such that $d\omega(p)=0$. This is for my homework of integration on manifolds & Stokes ...
6
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1answer
291 views

Integral of the ratio of two exponential sums

I am trying to find a lower bound on the following integral \begin{align*} \int_{y=-\infty}^{y=\infty} \frac{ (\sum_{n=[-N..N]/\{0\}}n e^{-\frac{(y-cn)^2}{2}})^2} {\sum_{n=[-N..N]/\{0\}} ...
0
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0answers
16 views

Show that $\lim_{y\xrightarrow{nt}x}\int_{\Sigma}f(y-z)g(z)d\sigma(z)=\lim_{\epsilon\searrow 0}\int_{|x-z|>\epsilon}f(x-z)g(z)d\sigma(z)$,

Show that $\lim_{y\xrightarrow{nt}x}\int_{\Sigma}f(y-z)g(z)d\sigma(z)=\lim_{\epsilon\searrow 0}\int_{|x-z|>\epsilon}f(x-z)g(z)d\sigma(z)$, $\forall g \in L^2({\sigma})$ here $x\in \Sigma$ $\Sigma$ ...
3
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3answers
55 views

Limit and Integral problem work verification-2

I have to calculate the following: $$\large\lim_{x \to \infty}\left(\frac {\displaystyle\int\limits_{x^{2}}^{2x}t^{4}e^{t^{2}}dt}{e^{x}-1-x - \frac{x^2}{2}- \frac{x^3}{6}-\frac{x^4}{24}}\right)$$ ...
0
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1answer
39 views

For what values is this integral convergent?

How can I find for what values of $r$ $$\int_0^\infty x^re^{-x}dx$$ converges? I started by rewriting it as $$\lim_{b\to\infty}\int_0^bx^re^{-x}dx$$ but am not sure how to figure it out from here.
2
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5answers
82 views

Finding limits using definite integrals $\lim_{n\to\infty}\sum^n_{n=1}\frac 1 n$

Find the limit of $\displaystyle\lim_{n\to\infty}\frac 1 {n^5}(1^4+2^4...+n^4)$ using definite integrals. It's equal to: $\displaystyle\lim_{n\to\infty} \sum^n_{i=1}\frac 1 i$ but now I'm not ...
4
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0answers
58 views

Integration Around Part of a Branch Cut

I am studying the integral, given by a Laplace transform, $$\int_0^\infty\!e^{-\alpha x}\sinh^{-2/3}x\left(1+\frac 12\sinh^2x\right)^{-1/6}\left(1-\beta\sinh^{4/3}x\right)^{1/2}\,\mathrm dx$$ From ...
1
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2answers
35 views

Integrating $ e^{3x^2} $

Let Ω be the region enclosed by the $x–axis$, the line $y = 2x$ and the line $x = 1$. Calculate $\iint_{\Omega }^{} e^{3x^2} dxdy $ I simply rewrote this integral as $ \int_{0}^{2} ...
1
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1answer
58 views

$\int^1_0(1-x)^8x^{11}-(1-x)^{11}x^8dx$ and replacing variables

Find $$\int^1_0(1-x)^8x^{11}-(1-x)^{11}x^8dx$$ I have an answer here but I don't understand the last part of it: ...
6
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0answers
101 views

How to evaluate the integral $\int_0^\infty \frac{x^{a-1}}{1+bx^a} e^{-x} dx$

How to evaluate this integral? \begin{equation} \int_0^\infty \frac{x^{a-1}}{1+bx^a} e^{-x} dx \end{equation} I think it will use a gamma function or a exponential integral. I really need an ...
0
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1answer
34 views

Proving for non-holomorphic function $g(z)$ that $\int _ \gamma g(z)dz=0$

Let there be function $g(z)=(z+3)(\overline z-3)e^{z^2}$. The curve $\gamma$ is a circle with radius $4$ and centered at $z=3$. Prove that $\int _ \gamma g(z)dz=0$. Now I can't use Cauchy theorem ...
0
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0answers
23 views

How to do integration with non-square matrix?

I am not sure how to do the following integration with a non-square matrix $A$. By Google search I could get results only for square matrix. But for non-square matrix $\det(A)$ does not exist. So how ...
0
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1answer
25 views

Limit of sequence and integral 5

I have to calculate $\lim_{n \to \infty}{\int_n^{n+7}{\frac{\sin(x)}{x}}}$. I guess $0 \le \int_n^{n+7}{\frac{\sin(x)}{x}} \le \frac{sin(n+7)}{n+7}$ so using the sqeeze theorem the answer is $0$. Am ...
0
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1answer
10 views

Calculating the inverse of a continuous map for a certain interval in order to calculate the Perron-Frobenius operator.

Suppose we are observing chaotic continuos maps, the Perron-Frobenius operator $P$ satisfies: $P\phi_{n}(t) = \frac{d}{dt} \int_{f^{-1}([a,t])} \phi(x)dx$ I don't understand how for the shift map, ...
2
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0answers
22 views

Indefinite Hypergeometric Integral Transformations

I'm attempting to solve the indefinite integral $$S\left(v\right) = 2a\sqrt{\alpha ...
2
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1answer
27 views

Integral and derivative

Let $g(x) = \int_{[0;2^x]}{\sin(t^2)} dt$ for $x \in \mathbb{R}$. I have to calculate $g'(0)$. So, $g'(0) = \lim_{h \to 0}{\frac{g(h) - \int_{[0;1]}{\sin(t^2)} dt}{h}}$. Maybe I should apply the ...
2
votes
3answers
90 views

Finding $\int ^\pi_{-\pi}\sin(nx)\cos(mx)dx$

Find $$\int ^\pi_{-\pi}\sin(nx)\cos(mx)dx$$ I used the product identity and got: $\displaystyle \int ^\pi_{-\pi}\sin(nx)\cos(mx)dx = ...
-1
votes
4answers
48 views

Limit of sequence and integral [closed]

How to calculate the following limit? $$ \lim_{x \to o^+}{\frac{\int_{x^3}^{7x^2}{\cos(t^{2010})} \mathrm{d}t}{x^2}} $$
1
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1answer
38 views

Integral of matrices

I want to perform the following integration: $$ \int_0^{dt} e^{-\kappa s}\Sigma\Sigma^T e^{-\kappa^T s} ds $$ where $\kappa$ is a $5\times5$ matrix as well as $\Sigma$. Is there any way to solve this ...
0
votes
1answer
26 views

Sum of integrals with variables shifted in each sum: How to justify this expression?

I annoyingly can't justify a step in the solution of the following problem. I have the following expression at hand: $$ \sum_{n=1}^{N}\int_{-\infty}^{\infty}{(y(x_n + \xi) - t_n})\nu(\xi)\eta(x_n + ...
0
votes
2answers
39 views

How to integrate $\int \frac{v}{g-kv}dv$

How to integrate $$\int \frac{v}{g-kv}dv$$ it's supposed to be equal to $$ -\frac{v}{k} -\frac{g}{k^2}ln(g-kv)$$ but I can't get that, I tried a substitution $u=g-kv$ and got close but no right? Any ...
2
votes
3answers
74 views

How evaluate $\int \frac{\cos^2(x)}{1 + \text{e}^x}dx$ to find an improper integral

Can someone help me evaluate this: $$\int \frac{\cos^2(x)}{1 + \text{e}^x}dx\;?$$ I need it for determining whether the improper integral $\int_0^\infty {\frac{{\cos^2{{(x)}}}}{{1 + ...
5
votes
3answers
80 views

Definite integral of $\cos (x)/ \sqrt{x}$?

How do I integrate $$ \int^{+\infty}_0 \frac{\cos(x)}{\sqrt{x}} dx$$ I tried setting $u = x^{-1/2}$ and $dv = \cos(x)dx$. Then I integrate by part twice to get: $$ \int^\infty_0 \frac{\cos ...
1
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2answers
37 views

Solving a triple integral with Spherical Coordinates

I am attempting to solve this integral $\iiint \frac{1}{\sqrt{x^2+y^2+(z-2)^2)}} \mathrm{dV}$ where the region $v$ is a unit sphere. How would I go about converting the function inside the integral ...