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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0
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1answer
78 views

Differentiation of multivariable function proof

I'm looking for the differentiation of multivariable function integral $$\frac{\mathrm{d} }{\mathrm{d} x} \int_{v(x)}^{u(x)}f(t,x)dt=u'(x)f(u(x),x)-v'(x)f(v(x),x)+\int_{v(x)}^{u(x)}\frac{\partial ...
3
votes
2answers
63 views

Where is my mistake $\iint_{Q} (x+y)^{2013}dxdy$

I'm preparing for a calculus exam, and I tried to solve the following question. $Q$ is square $[-1,1]^2 \subset \mathbb R^2$ We are asked to evaluate $\iint_Q (x+y)^{2013}dxdy$ Here is what I did: ...
1
vote
2answers
115 views

Using series find $\int_0^1 \sqrt{1+x^4}\hspace{1mm} dx$ up to $2$ decimal places

I cannot figure out an aesthetic way to do this. Can someone give a beautiful solution to this ugly question? This is what I have tried yet. I used the fact that $$x = ...
3
votes
1answer
75 views

How to prove this integral inequality?

Here is a problem: Let $B_r=\{ (x_1,x_2,\cdots,x_n)\in \mathbb{R}^n: x_1^2+x_2^2+\cdots+x_n^2<r^2\}.$ Let $f$ be a $C^1$ real function on $B_2$. Prove that $$\inf_{a\in R}\int_{B_2} ...
1
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2answers
69 views

Help in understanding the Maple answer

I tried to find the definite integral of an equation using Maple. It returns the answer to me as a limit. Specifically, it is giving me $$\lim_{r\to \infty} G(r)$$ What does this answer mean? Do I ...
6
votes
2answers
260 views

Integral: $\int_0^\infty \tan^{-1}\left(\frac{2ax}{x^2+c^2} \right)\sin(bx) \; dx$

Please help me in proving the following result: $$\displaystyle \int_0^\infty \tan^{-1}\left(\frac{2ax}{x^2+c^2} \right)\sin(bx) \; dx=\frac{\pi}{b}e^{-b\sqrt{a^2+c^2}}\sinh (ab)$$ I found this ...
0
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1answer
38 views

how to show function f is integrable or not

Here if $f$ is a measurable function on [0,1], define $G(x,y)$ $$G(x,y)=f(x)-f(y),$$ If we know $ G(x,y)\in L^1([0,1]\times[0,1])$, then $f$ is integrable on $[0,1]$ or not? If not, can we give a ...
0
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1answer
59 views

A question on integration by parts

From PDE Evans, 2nd edition, page 148... Observe \begin{align} \int_0^\infty \int_{-\infty}^\infty w_t v_x \, dx dt &= -\int_0^\infty \int_{-\infty}^\infty wv_{tx} \, dx dt - ...
2
votes
1answer
45 views

Calculate $\int_D x^3y\ dx\,dy$

Let $D$ the bounded region by the $y$-axis and the parable $x= -4y^2 + 3$. How can I calculate the integral $$\int_D x^3y\ dx\,dy$$ I am stuck with this problem some help to solve this please.
12
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1answer
223 views

Proving that $\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$

How can we prove, without employing the aid of residues or various transforms, that, for $n>2$ $$\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$$ Motivation: ...
5
votes
0answers
69 views

An inequality between integrals of series of characteristic functions of cubes

Let $1\leq p<\infty$. Prove that there exists $C>0$ such that $$ \left(\int\left|\sum_{i=1}^\infty a_i\chi_{2Q_i}\right|^p \, dx\right)^{1/p} \leq C\left(\int\left|\sum_{i=1}^\infty ...
11
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1answer
175 views

The integral on $[0,1]\times[0,1]$

Here I have a problem. $p$ and $q$ are positive numbers. the integral $$\int_0^1\int_0^1 \frac{1}{x^p+y^q}\;dx\;dy< \infty \Longleftrightarrow \frac{1}{p}+\frac{1}{q}>1$$ Here is my try, ...
0
votes
0answers
42 views

Integral of different types of noise

I have been learning about the Wiener process and read this on Wikipedia; "the Wiener process is used to represent the integral of a Gaussian white noise process" It got me wondering what the ...
0
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1answer
44 views

Double integral to a single integral?

Let $P(t)$ is a function $\mathbb{R}\rightarrow \mathbb{R}^{m\times m}$ and $t$ is its variable. Give that $P(t_1+t_2)=P(t_1)+P(t_2)$. $\int_0^\infty\int_0^\infty C P(t_1)P(t_2) dt_1 ...
0
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2answers
59 views

$\int_0^\pi\sin(2t)e^{-in2t}dt$ complex number integral for integer values of n

$$\int_0^\pi\sin(2t)e^{-in2t} \, dt$$ wolfram alpha say the answer is $$\frac{1-e^{-2 i n π}}{2-2 n^2}$$ although using the integral trig identity $$\int ...
1
vote
2answers
55 views

Integrating a differential form inside a cylinder

Let S be cylinder given by $x^2+y^2=1$ between $z=1$ and $z=3.$ For $\varphi=e^xdx\wedge dy+ ydz\wedge dx+xdy\wedge dz$ find $\int_S\varphi$. I managed to finish the problem, but I'm getting ...
2
votes
2answers
187 views

Inverse Laplace with $\ln$

How can I compute the inverse Laplace of 1) $\ln\left(\dfrac{s+1}{s-1}\right)$ 2) $\ln\left(\dfrac{s-1}{s}\right)$. Can someone please hep me to do these two problems
3
votes
2answers
109 views

Evaluate by contour integration $\int_0^1\frac{dx}{(x^2-x^3)^{1/3}}$

Evaluate by contour integration [i am learning complex analysis - calculus of residues] $$\int_0^1\frac{dx}{(x^2-x^3)^{1/3}}$$ I tried by taking $x^3$ out from the denominator but that didnt work.
1
vote
1answer
56 views

Does $\displaystyle\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) dx\,$?

If $\displaystyle\{f_n(x)\}$ is sequence of continuous function on $\mathbb{R}$ converging uniformly to $f(x)$,then does $\displaystyle\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = ...
2
votes
4answers
100 views

integration by parts of trig functions

Can anyone help me with this integral? $\int{x^3 \sin(x^4) dx}$ I set $u=x^3$, and I let $v=-\cos(x^4)$, so that $\frac{dv}{dx}=\sin(x^4)$ I tried using integration by parts, but, whenever I come ...
4
votes
3answers
200 views

Integration by change the variable

Let, $\int_{-1}^1\sqrt{1+e^x}\operatorname{dx}$. Write as an integral of a rational function and compute it. Suggest: change the variable in order to eliminate the square root. My work was: ...
7
votes
5answers
363 views

How find this limit $I=\lim_{n\to\infty}n^a\left(\int_{0}^{\pi/2}\sin{(nx)}\cos^n{x}dx\right)=b$

If the constant $a,b\neq 0$ such $$ I=\lim_{n \to \infty}\left[% n^{a}\int_{0}^{\pi/2}\sin\left(nx\right)\cos^{n}\left(x\right)\,{\rm d}x \right] = b $$ find $a,b$ My idea: since ...
0
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3answers
50 views

Supposedly simple integration question

It has been a fair while since I've done any mathematics (took a semester off after the long summer break), and therefore maybe I am missing something obvious, but I cannot seem to solve the ...
1
vote
1answer
41 views

Why is $(\sec x)' = \tan x\sec x$ and not $\tan x$?

As far as I understood, the Fundamental Theorem of Calculus states that the integral of a function is its anti-derivative. And yet, although the integral of $\tan x$ is $\sec x$, the derivative of ...
0
votes
1answer
31 views

Positive continuous function with bounded differential $\int_{0}^{\infty}{f}<\infty$ tends to zero

Could you help me with this a little? Let $f$ be a continuous function s.t. $f' \leq C$, $f \geq 0$ and $\int_{0}^{\infty}{f}<\infty$ then $\lim_{t \to \infty}{f(t)}=0$. What I found so far: I ...
0
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4answers
47 views

derivative $\frac{df(t)}{dt}$ of $f(t) = \int_0^t\ln{(s^2+t^2)} ds$

Let $f(t) = \int_0^t \ln{(s^2+t^2)} ds$, how can I find the derivative $\frac{df(t)}{dt}$? The function $\,\int_0^t \ln{(s^2+t^2)} ds$ is defined to be continuous in $s^2+t^2 > 0$ and $ s^2+t^2 ...
4
votes
5answers
176 views

Evaluate $\int \sqrt{1-x^2}\,dx$

I have a question to calculate the indefinite integral: $$\int \sqrt{1-x^2} dx $$ using trigonometric substitution. Using the substitution $ u=\sin x $ and $du =\cos x\,dx $, the integral becomes: ...
0
votes
1answer
41 views

Evaluating one integral in terms of another

If $$S=\int_0^1\frac{e^t}{t+1}dt$$then what is the value of :$$\int_{a-1}^a\frac{e^{-t}}{t-a-1}dt$$ The answer is given as $-Se^{-a}$
0
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0answers
88 views

Solving integral that contain upper incomplete gamma function, exponential, and powers

I have this integration formula; $ f=\int\limits_{0}^{\infty}\frac{e^{-b~z}}{\sqrt{z}} \Big(\frac{\beta}{\beta+z}\Big)\Big(\frac{\beta+z}{z}\Big)^L ...
1
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0answers
29 views

When can we use Gauss Hermite integration technique

I want to evaluate the following integral:$\\ \int_{-\infty}^{\infty} e^{-\frac{x^2}{2}}e^{f\left(x\right)}\frac{dx}{x}$. Where, $f\left(x\right)$ is a function of $x$. Is it correct to evaluate the ...
2
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1answer
51 views

Flux Integral - where did I go wrong?

S is the graph $z=25-(x^2+y^2)$ over the disk $x^2+y^2\leq 9$ and $\varphi = z^2dx\wedge dy$. Find $\int_S \varphi$. According to the book the answer is $3843\pi$, but the answer I got is ...
0
votes
2answers
60 views

Double integral - Volume

I find it very difficult to solve this problem. I need help setting the integral up. I know that the first one is cone and the third one is paraboloid, but I can't define the limits. Some explanation ...
1
vote
1answer
35 views

One dimensional integrals in Green's theorem

I am trying to understand Green's theorem, but the problem is I don't know what is the definition of the integrals in the theorem. This is the expression that one proves to hold with some assumption ...
1
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2answers
119 views

Contour integral $\int^\pi_{-\pi}(a-\cos\theta)^b\exp(c\cos\theta)d\theta$ assuming $a>1$, $b>0$, $c>0$

Under the condition $a>1$, $b>0$, $c>0$, is there any good function to express the following integral? $$ \int^\pi_{-\pi}\left(a-\cos\theta\right)^b\exp\left(c\cos\theta\right)d\theta $$ I ...
0
votes
2answers
47 views

Need help calculating the integral $\iint_{\mathbb R^2}e^{-(|2x-y| + |2y-x|)}\,dx\,dy$

$$ \text{I'm asked to calculate}\quad \iint_{{\mathbb R}^{2}} {\rm e}^{-\left\vert\,2x - y\,\right\vert\ -\ \left\vert\,2y - x\,\right\vert} \,{\rm d}x\,{\rm d}y. $$ I've substituted $u = 2x-y, v = ...
1
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1answer
41 views

Triangle inequality for integrals with complex valued integrand

This is a step in a lecture note I'm reading. It should be simple because the author considers it obvious but I can't see it. What am I missing? Suppose $U$ and $V$ are integrable over measure space ...
2
votes
1answer
32 views

weighted integral in convex hull

Working on an integral $$ J=\frac1{2\pi} \int_0^{2\pi} w(t) g(e^{it}) dt $$ where $\frac1{2\pi} \int_0^{2\pi} w(t) dt=1$ ; $w(t)$ is non-negative continuous ...
0
votes
0answers
19 views

closed form of integral of special function? $\int_{0}^{1} e^{a\, q^{1/k}\, {}_1 F_{2}(1,1/k,1 + 1/k;q)} d q$

Take the following integral, defined by hypergeometric functions: $$\int_{0}^{1} e^{a\, q^{1/k}\, {}_1 F_{2}(1,1/k,1 + 1/k;q)}d q$$ (there is a similar formulation Lerch). I think the series ...
1
vote
1answer
115 views

showing $\int _a^b\left(f'\left(x\right)\right)dx\:=\:f\left(b\right)-f\left(a\right)$

Let $f(x):[a,b]\to \mathbb R$, be differentiable on $[a,b]$ (and continuous) so that $f'(x)$ is integrable on $[a,b]$. I need to show that: $$\int _a^b\left(f'\left(x\right)\right)\mathrm dx = ...
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votes
0answers
20 views

Calculate the weights and the node in the integration formula

The problem is the following. Calculate the weights $w_1$ and $w_2$ and the node $x_1$ in the weighted integration formula $\int_0^1x^{\frac{3}{4}}f(x)dx\approx w_1f(x_1)+w_2f(\frac{3}{4})$ The ...
1
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1answer
56 views

Finding the limit of an integral

Evaluate $$\displaystyle\lim_{j\rightarrow \infty} \displaystyle\int_{0}^{a} \frac{1}{j!} \left(\ln \left(\frac{A}{x}\right)\right)^{j}dx$$
0
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0answers
14 views

Expected value $E[(q-D_1-D_2)^+]$

I have two demand variables $D_1$ and $D_2$ and a given variable $q$. We know that $D_1 \sim U(0,a)$ and that $D_2 = \lambda - \alpha p + \epsilon$ where $\epsilon \sim U(0,b)$. Let $f_1(x)$ denote ...
0
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1answer
48 views

Can't figure out a way to integrate $\int_{A}\log(\sin(x-y))dx dy$

I'm having trouble computing the integral $\int_{A}\log(\sin(x-y))dx dy$, where $A$ is the triangle defined by the 3 eqations: $x = \pi, y = x ,y = 0$ I tried the substition $u = x-y, v = x+y$ and ...
2
votes
2answers
120 views

Monte Carlo Importance Sampling

I am reading the book on Monte Carlo by Sobol (A Primer for the Monte Carlo Method). In the section on Importance Sampling, he writes: $I = \int_a^b g(x) \: dx$ "to compute this integral, we could ...
6
votes
3answers
555 views

How do you integrate the reciprocal of square root of cosine?

I encountered this integral in physics and got stuck. $$\int_{0}^{\Large\frac{\pi}{2}} \dfrac{d\theta}{\sqrt{\cos \theta}}.$$
0
votes
1answer
69 views

Integration in Maple

I am trying to evaluate the following integral using maple but it returns the answer without evaluating the lower limit. Can anybody help me in using maple to solve this integral? $$ ...
4
votes
0answers
87 views

Evaluate $\int \dfrac {x^2}{\sqrt{\arctan x}} dx$

Is there any closed form expression of $$\int \dfrac {x^2}{\sqrt{\arctan x}} dx?$$
0
votes
2answers
32 views

$\int\exp(-jnw_0t)\,dt$ integral calculus.

I seem to forgot these parts of integral calculus. I am trying to determine the Fourier coefficient in complex exponential form. Here, $t$ is the variable being integrated and $n$ is for all ...
0
votes
1answer
51 views

Evaluating $\int_A y^2 dA$

I would like some explanation regarding the following integral which is not familiar to me: $$\int_A y^2 dA$$ where $y$ could be a function of $A$ and $A$ is cross section. It's named $I_z$ in the ...
0
votes
1answer
26 views

Comparing Chebychev's inequality to the exact probability

Let $X$ be continuous with pdf $f(x)=e^{-x}$ if $0<x<\infty$, and zero elsewhere. $(1)$ Use Chebychev's inequality to obtain a lower bound on $P(-1.5<x<3.5)$ Here's what I did: ...