Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

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0answers
54 views

Asymptotic evaluation from Bender and Orszag

In Bender and Orszag's Advanced Mathematical Methods for Scientists and Engineers, Exercise 6.56 (c) asks to find the leading behaviour of the integral $ \int_0^1 \cos(x t^4)\tan t\;\mathrm{d}t$ as $x\...
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1answer
69 views

$\int\sqrt{\sin(x)}(\cos(x))^3dx$

EDIT: @Andrew Chin's comment solved the issue $\int(\sin(x))^{1/2}(\cos(x))^3dx$ =$\int(\sin(x))^{1/2}(\cos(x))^2(\cos(x))dx$ =$\int(\sin(x))^{1/2}(1-\sin^2(x))(\cos(x))dx$ Let $u=\sin(x)$, therefore $...
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2answers
61 views

How to compute a complex integral of $x \: dz$?

I'm working through chapter 4 of Alfohrs, and the first exercise is computing the integral $\int_\gamma x \: dz$ where $\gamma$ is the directed line segment from $0$ to $1+i$. What, exactly, does this ...
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1answer
38 views

discrete sum greater than integration

Why for any $j \in N$ (the natural numbers) we have $\sum_{n = j+1}^{j^2} \frac{1}{n} \geq \int_{j+1}^{j^2 + 1} \frac{1}{x} dx $
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3answers
146 views

A different way to calculate $\int_0^\infty \tfrac{1}{1+x^n} dx$.

Most of you might have stumbled upon this integral, when studying complex analysis, namely via contour integration, but today I asked myself if there is a different way to calculate it. I'm almost ...
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55 views

How many points of integration are necessary to approx $\int_0^1{e^{e^{-x}}}dx$ with 4 correct decimal places using the trapezoid rule?

How many points of integration are necessary to approx $\int_0^1{e^{e^{-x}}}dx$ with 4 correct decimal places using the trapezoid rule? I know that what I need here is $$|e_r| \le 5\times 10^{-(m+1)},...
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1answer
36 views

What to set $u$ as in integration by substitution

$$\int_{\frac{1}{2}}^{\frac{\sqrt{3}}{2}}x^2\sqrt{1-x^2}\ \mathrm{d}x$$ The question asks me to use a trigonometric substitution so I tried $u=\sin x$ but that doesn't give me nice $x$ limit values ...
4
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2answers
92 views

$ \int_c^\infty \frac{e^{-a x^2 + bx}}{x} \, dx $

Suppose $a > 0$, $b$ is an arbitrary real number, and $c$ is a positive real number. Is there any closed form solution for the following integral $$ \int_c^\infty \frac{e^{-a x^2 + bx}}{x} \, dx \...
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2answers
72 views

What is the antiderivative of one over an arbitrary polynomial?

Suppose we have an arbitrary polynomial as follows. $$p(x):=\sum_{i=0}^n a_nx^n$$ And we want to find the antiderivative of $(p(x))^{-1}$ with respect to $x$. $$\int\frac{dx}{p(x)}$$ Messing around ...
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4answers
79 views

How can we simplify this integral? $\int{x\frac{f(x)}{\int f(x) dx} dx}$

In a machine learning lecture, we encountered the following integral that we needed to solve to calculate the mean of some random variable $x$: \begin{equation*} \int{x\frac{f(x)}{\int f(x) dx} dx} \...
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1answer
54 views

Calculate $\iint_D \frac{\sqrt{x^2 + y^2}}{1+x^2+y^2} dx\,dy$

I am attempting to solve the integral: $$\iint_D \frac{\sqrt{x^2 + y^2}}{1+x^2+y^2} dx\,dy\,,$$ where $D$ is bounded by $1<x^2+y^2<9$ and within the sector bounded by lines $$-\frac{y\sqrt{2}}{2}...
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27 views

2D Fourier Transform of a function?

Here I have a function like: $\phi(x,0,z) = \int_0^\infty\int_0^\infty A_1\sin(ux)\sin(vy)dudv$ and I would like to find the form of $A_1$ which I have been told is the 2d Fourier Transform of $\phi(x,...
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0answers
28 views

$f: [0, 1] \to \mathbb{R}$ is continuous. Prove existence of $c \in (0, 1)$ where $\int_{0}^{1} f(x) x^2 dx = \frac{f(c)}{3}$ [duplicate]

My approach with partial integration: $$\int_{0}^{1} f(x) x^2 dx = \frac{f(1)}{3} - \int_{0}^{1} f'(x) x^2 dx = \frac{f(c)}{3}$$ By mean value theorems, there exists $d \in [0, 1]$ such that: $$\int_{...
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60 views

Calculating $p$-norms for $f_\epsilon(x)=x^{-1/p+\epsilon}\chi_{(0,a)}(x)$

I have some problems calculating this norm, I can't get the integral right. I have the following function $f_{\epsilon}(x)= x^{(-1/p)+\epsilon}\mathcal{X}_{(0,a)}(x)$ with $p>1$, $a>0$ and $0&...
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48 views

Need to solve an integral [closed]

I need help to solve the following integral $\int_{0}^{\infty}x^n \exp(-ax^m-bx^2)\,dx$ Any help will be appreciated.
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1answer
23 views

Using the surface of the unit sphere to calculate $\int_{\mathbb R^3}\chi_{\{x_3>0\}}\chi_{B^c_{1/n}(0)}\exp(-\lvert x\rvert^2)/\lvert x\rvert dx$

I'm struggling with finding a proof idea for this integral from an integration theory practice exam. Suppose $$f_n:=\frac{\exp(-\lVert x\rVert_2^2)}{\lVert x\rVert_2}\chi_{\{x_3>0\}}(x)\chi_{B^c_{1/...
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2answers
103 views

Convergence of integral and summation for Time taken for complete revolution around vertical circle

I want to find Time taken to complete Vertical circular motion by Particle of mass $m$ So I proceed as follows Applying work energy theorem $$-mgR(1-\cos(\theta)=\frac12 mv^2-\frac12 mu^2$$ $$\implies ...
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2answers
70 views

Solving integral involving Dirac delta function i.e$\int_{-\infty}^\infty e ^{\delta(x)} dx$

I know some properties of Dirac-delta function specially those which are useful for integration involving these functions such as $$\int\limits_{-\infty}^{\infty}\delta(x)\,\mathrm{d}x$$ But I am ...
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1answer
31 views

Euler's Beta function for positive integers derivation

I am aware that a proof for the Euler's Beta function for positive integers has already been asked, but I have tried to derive it myself without using the gamma function, and my result doesn't match. ...
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0answers
28 views

What tools can we use to visualise volume integrations?

What tools do people use to visualize integrated volumes? I'm horrible at drawing perspectives, especially with hollowed-out 3D graphs. Can't imagine how to do that going beyond the usual three ...
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2answers
68 views

Integral involving exponentials of cosh functions

I recently encountered the following integral $$I_p\equiv \int\limits_0^{+\infty}\,\int\limits_{-\infty}^{+\infty}\,e^{-p\,[c+(c^2+1)\cosh x]}\,\mathrm{d}x\,\mathrm{d}c\,,$$ with $p>0$. Can $I_p$ ...
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0answers
28 views

Integral approximation (Laplace's method)

in order to show a result relating to Laplace's method of approximating integrals, I am stuck with the following exercise. Let $f,g\in C(I,\mathbb R)$ where $I$ is a compact interval whose interior ...
1
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1answer
68 views

Area: Shell of a simple spiral [closed]

Staying inside the math of $2$D Polar coordinates, up to but may not include formulas for areas. The general equation is $r=f(θ)$ for many types of spirals. A point in polar coordinates is $(r,θ)$. ...
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1answer
91 views

Convergence integral Riemann zeta

In the paper of Riemann and the book of Edwards I encountered the following representation for $\zeta$: $$ \int_0^\infty\frac{x^{s-1}}{e^x-1}\,dx=\Pi(s-1)\zeta(s) $$ which due to the fast growth of $e^...
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1answer
142 views

Prove $\int_0^1 f(g(x)) \, dx \leq \int_0^1 f(x)\,dx + \int_0^1 g(x)\,dx$

$f$ and $g$ are continuous functions on $[0, 1]$. $f$ doesn't decrease on this segment. Values of both $f$ and $g$ are in range $[0, 1]$. Prove following inequality $$\int_0^1 f(g(x)) \, dx \leq \...
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3answers
60 views

Integrate $\int_2^5 \dfrac{1}{1+\sqrt{x-1}}\, \mathrm{d}x $ using substitution with a fraction

Im doing this question currently $$\int_2^5 \dfrac{1}{1+\sqrt{x-1}}\, \mathrm{d}x $$ and I set $u = x-1$ to get $\tfrac{\mathrm{d}u}{\mathrm{d}x} =1$ so $du =dx$. I then write it out as $$\frac{1}{1+\...
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0answers
24 views

Ratio of Definite Integrals

I have ended up with the following ratio of definite integrals: $$g(z)=\left[{\int\limits_0^1 f(x)e^{-zf(x)}dx}\right]^{-1}{\int\limits_0^1 f^2(x)e^{-zf(x)}dx}$$ where $f(x)$ is an unknown function. ...
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0answers
25 views

Counterexample to: $f$ is Riemann Integrable if and only if its discontinuous set is Jordan measure zero

The forward direction is covered in some textbooks. But the reverse I am only able to find sparingly. Is this a counterexample?: $f=\begin{cases} 0 & (x\not\in\mathbb{Q})\\ 1/n & (x\in\mathbb{...
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3answers
61 views

Integration with trig substitution

Trying to evaluate this using trig substitution: $$\int \frac {1}{49x^2 + 25}\mathrm{d}x $$ Here's how I'm going about it, using $x = 5/7(\tan\theta)$ $$\int \frac {1}{49\left(x^2 + \frac{25}{49}\...
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1answer
34 views

For $f(t)=t^2-4t+3$ find $g(x)=\min_{(x\le t\le x+1)} f(t)$ and a value depending on $g$ [closed]

Let $f:\mathbb{R}\to\mathbb{R}$ $ ,f(t)=t^2-4t+3$ and $g(x)=\min_{(x\le t\le x+1)} f(t)$ . Find $$a=\frac{1}{ g'(1/2)+g'(3/2)}\int\limits^3_0 g(x)\,\mathrm{d}x$$ I had this problem on a test today and ...
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1answer
50 views

Unable to solve definite integral if an implicit function

We were asked to find the area under the given function, as a challenging exercise: \begin{gather*} 6x^{3} +11x^{2} y+6xy^{2} +y^{3} =x \end{gather*} The first thing I tried to do, was probably naive, ...
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2answers
31 views

Definite Integral from or to; plus or minus infinity

I always knew integrating in a common way: $\int_{n}^{z}{x^y} \ dx=[\frac{z^{y+1}}{y+1}] - [\frac{n^{y+1}}{y+1}]$ All we do is integrate $x^y$ and then replace the $x$ with $z$ first, then subtract it ...
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0answers
55 views

Integration by parts to form an infinite series

The integral $$I(\theta)=\int \sin(\theta)\frac{d\theta}{dt}dt$$ can be evaluated with the quotient rule by writing $I(\theta)=\int \sin(\theta)d\theta $. Instead, apply integration by parts more than ...
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0answers
17 views

Monotonic and non monotonic function's integrability(Riemann) question

I was told that all functions that have a finite number of discontinuities, are Riemann-integrable. Then, I found a proposition that tells the following: All monotonic functions on closed interval $[a,...
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2answers
108 views

Hard improper integral $\int_0^\infty \frac{(1+\frac{1}{x})^{-x}}{x^2}dx $

I have been stuck on this problem for a while: $$\int_0^\infty \frac{(1+\frac{1}{x})^{-x}}{x^2}dx $$ I have toyed around with substitution and integration by parts with no luck. I have also searched ...
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0answers
29 views

$\frac{d}{dx} \int_{x_0}^{x_1} \rho u_{tt} dx$

In Partial Differential Equations by Walter Strauss, Ch 1.3 Example 2, the author differentiates the integral $$\int_{x_0}^{x_1} \rho u_{tt} dx$$ with respect to $x$ and finds that; $$\frac{d}{dx} \...
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2answers
52 views

Calculate polygon using some integrals

I have a polygon below. The length of it is $a$, base is $b$, and height is $h$. I want to have an integral to calculate volume of it. This is so different from what I know before, and it's not graph ...
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1answer
30 views

Argument $\int_{-\infty}^{t}\int_{-\infty}^{x_2}f_{X_1,X_2}dx_1dx_2=\int_{-\infty}^{t}\int_{-\infty}^{x_1}f_{X_1,X_2}dx_2dx_1$

Right now I have two integrals $$ \int_{-\infty}^{t} \int_{-\infty}^{x_2} f_{X_1,X_2}(x_1,x_2) \, dx_1 dx_2 = \int_{-\infty}^{t} \int_{-\infty}^{x_1} f_{X_1,X_2}(x_1,x_2) \, dx_2 dx_1 $$ with the ...
2
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3answers
133 views

Prove that : $ \int_{0}^{2\ln{\varphi}}{\theta\ln{\left(2\sinh{\frac{\theta}{2}}\right)}\,\mathrm{d}\theta}=-\frac{1}{5}\zeta\left(3\right) $

Denoting $ \varphi=\frac{1+\sqrt{5}}{2}=\mathrm{Golden\ Ratio} $. How would you prove that : $$ \int_{0}^{2\ln{\varphi}}{\theta\ln{\left(2\sinh{\frac{\theta}{2}}\right)}\,\mathrm{d}\theta}=-\frac{1}{...
1
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1answer
50 views

An Idea to solve a limit involve integral [duplicate]

Suppose $f(x)$ is a continuous function over $[0,1]$,and $0<a<b$ find the below limit $$\lim_{t\to 0^+}{\int_{at}^{bt}\frac{f(x)}{x}}=?$$ My work is below $\int_{at}^{bt}\frac{f(x)}{x}=\int_{at}^...
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1answer
61 views

Integral of two error functions times a Gaussian

I am trying to find some expression for this integral, ideally as a function of the constants. I can't put the full integral in the title as it's too long. I doubt a closed form exists, but I'm even ...
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0answers
21 views

Behaviour of the sum of a function of a random variable and its PDF

I have $\int_{R}^{\bar{v}}(R-c)f(v)dv$, where $f(v)$ is the pdf of $v\sim[\underline{v},\overline{v}]. $ I am trying to determine how this integral changes as $R$ changes. I understand that as $R$ ...
0
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0answers
17 views

Differentiating with respect to the lower bound of an integral with a PDF

I am trying to find $\frac{d}{dR}\int_{R}^{\bar{v}}(v-R)g(v)dv$, where $v$ is a random variable distributed on $[\underline{v}, \bar{v}]$, with pdf $g(v)$ and CDF $G(v)$. I found that: \begin{array}{l}...
1
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2answers
89 views

Step by step method for $\int\frac{x^2+1}{x^2-1}\,dx$

$$\int\frac{x^2+1}{x^2-1}\,dx$$ What is the step-by-step way of solving this integral problem? I tried using substitution which was $x^2-1=t^2$, but end up with an even more complicated equation. ...
3
votes
1answer
100 views

Compute the following double integral… [closed]

Prove that $$\int\limits_{0}^{1}\int\limits_{0}^{1}\dfrac{dxdy}{\sqrt{x(1-x)}\sqrt{y(1-y)}\sqrt{1-xy}}=\dfrac{1}{4\pi}\left(\int\limits_{0}^{\infty}e^{-t}t^{-3/4}dt\right)^4$$ Please give some hint/...
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4answers
63 views

Find $c\in\mathbb{R}$, and $f$ such that $\int_c^xt\cdot f(t)\,\mathrm{d}x=x\sin (x)+\cos (x)+\frac{x^3}3$

Find a number and a function that satisfies: $$\displaystyle\int_{c}^{x}t\cdot f(t)\,\mathrm{d}x=x\sin (x)+\cos (x)+\dfrac{x^3}{3}$$ I tried taking derivative both sides so I got: $$\left(\...
1
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1answer
47 views

How to prove that $\int_{\partial A}\frac{x}{\lVert x\rVert^n}\cdot \nu(x)d\sigma = \int_{\partial B_1(0)}1d\sigma$?

I am having some difficulty finding a proof idea for this question I found in an old exam: Suppose $A\subset\mathbb R^n$ is a bounded $C^1$ set with outer unit normal $\nu_A(x)$ and $0\in A$. Prove ...
2
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1answer
27 views

Variable substitution in double integral

Let $(a,b)\in (0,1)$ and let $T>0$. Consider the following integral: $$ \iint_{\Gamma }f(x)g(x+t)\,dt\,dx, $$ where $ \Gamma =\left\{ (t,x)\in (0,T)\times (0,1):t+x\in (0,1)\right\} . $ Let $s=x+t$...
2
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1answer
33 views

When can we affirm this equality about integral?

Suppose we have $f(x) = \int G(x_{0},x_{1},...)\,dx_{0}\,dx_{1}\dotsb dx_{n}$ When can we affirm that $$df = G(x_{0},x_{1},...)? \tag1$$ Basically, I am having trouble to understand how to deal with ...
1
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1answer
32 views

Calculate $\iint_D 2x*cos(2(x-y)x) 𝑑𝑥𝑑𝑦$ where D is a rectangle with corners at (π/6 ,2+ π /6 ), ( π/6,4+ π/6), (π /4,4+ π/4) and (π/4,2+ π/4)

I have been trying to calculate the double integral: $\iint_D 2x*cos(2(x-y)x) 𝑑𝑥𝑑𝑦$ where D is a rectangle with corners at (π/6 ,2+ π /6 ), ( π/6,4+ π/6), (π /4,4+ π/4) and (π/4,2+ π/4) for a ...