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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127 views

Find the approximations T4 and M4 and give error bounds.

a.Find the approximations T4 and M4 for Integral from 1 to 2 35e^(1/x) b. Estimate the errors in the approximations of part (a). (Round your answers to six decimal places.) For this park you use the ...
2
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2answers
66 views

How to integrate: $\frac{1}{(x^2 + z^2)^{\frac{-3}{2}}} dx$

I have tried to use u-substitution but for some reason am not doing it right and thus not getting the correct answer. I want to know the most obvious/ intuitive way to solve this integral.
3
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1answer
94 views

How do we know that $\int_0^1 x^n dx=\frac{1}{n+1}$?

What are some ways to prove $$\int_0^1 x^n dx=\frac{1}{n+1}$$ straight from (any) definition of the Riemann-/regulated integral? Or do we need the fundamental theorem of calculus and (anti)derivatives ...
3
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2answers
85 views

How to evaluate the limits of $\int (\sin y) dy$?

How to evaluate this integral of $$\int^{\arctan2}_{\arctan \frac{1}{2}} \sin y ~dy$$? Now I know this is $$-[\cos(\arctan 2)-\cos(\arctan 0.5)]=[\cos(\arctan 0.5)-\cos(\arctan 2)]$$ but I have no ...
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54 views

Name of multidimensional propagator integral

$$ I_{n,m}(\boldsymbol{x},\boldsymbol{\tau}) = \dfrac{1}{(2\pi)^{n+m}} \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \dfrac{e^{i(\boldsymbol{p}\cdot \boldsymbol{x}-\boldsymbol{q}\cdot ...
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1answer
38 views

Using the chain rule of differentiation to evaluate an integral along a curve

I have a little confusion regarding the following: $\gamma $ is a piecewise smooth curve from $A$ to $B$ and $h(x,y)$ is a continuously differentiable function on $\gamma$. Let ...
1
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1answer
42 views

Estimate of a (integral) function

I should show that function $H(w)=\int_{-\pi}^{\pi}f(x) e^{iwx}dx$, where $f(x)\in L^2(-\pi,\pi)$, is such that $H(re^{i\theta})=O(e^{\pi r |sin(\theta)|})$. Any suggestion?
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1answer
31 views

question over a integration changes order and hard to compute

I am trying to integrate the following function: $$f(x,y)=xy\times e^{-\frac{x^2y^2}{2}},x\in(1,2), y\in(0,\infty)$$ To do the integration: $$\int_1^2 \int_0^\infty ...
5
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2answers
101 views

Why is it legal to take the antiderivative of both sides of an equation?

first, I must apologize for somewhat misleading a title. To save both your and my time, I will go straight to the point. By definition, an indefinite integral, or a primitive, or an antiderivative ...
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1answer
33 views

Convergence of difference of series

Let $f(x)$ be a probability density function, i.e $f(.) \geq 0$ and $\int_{\mathbb{R}}f(x) \ dx=1$. Let $I_{n,k}= \big[\frac{k}{n}, \frac{k+1}{n}\big), \ k \in \mathbb{Z}$ and $p_{n,k}$ denote the ...
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1answer
29 views

How to find probability after finding the CDF of a max?

I have the pdf: $$ f(x)=10x^9, 0 < x < 1$$ Assuming independence, let Y=max(X1,...X8). Find the cdf for Y: I found this to be y80 Now I have to find P(.9999 < Y < 1) I thought this ...
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0answers
22 views

How to show that $\int_{\delta D} x\ dx $ is area of $D$

Prove that $\int_{\delta D}x\ dy$ is area of the $D$ and $\int_{\delta D}y\ dx$ is munis the area of $D.$ Now using Green's Theorem I can prove that $$\int_{\delta D}x\ ...
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0answers
134 views

Finding the value of a constant given a probability density function

So we're given the following information: Let X be a random variable with probability density function: $f(x)=c(1-x^2)$ if $-1<x<1$ $f(x)=0 $ otherwise Find the value of c How do I work ...
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0answers
23 views

Integrating using Green's Theorem

Evaluate $\int_{\gamma}xy\ dx $ where $\gamma$ is the boundary of the square with vertices $(0,0),(1,0),(1,1),(0,1)$. Now Green's Theorem says that $D$ be a bounded domain with piecewise ...
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1answer
38 views

Why is $\left|\int_0^{R_1} \frac{e^{-xR_2}\sin x}{x}dx \right| \leqslant \int_0^{R_1} e^{-xR_2}dx$ [closed]

Let $R_1 \in \mathbb{R^+}$ and $R_2 \in \mathbb{R^+\cup\{0\}}$ Since $|\sin x/x|<1$ for $x>0$, why, $$\displaystyle\left|\int_0^{R_1} \frac{e^{-xR_2}\sin x}{x}dx \right| \leqslant \int_0^{R_1} ...
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0answers
35 views

antiderivative of jump-discontinuous function

As I was teaching abroad, I came across a rather simple (if you don't think) problem that looks as follows: compute the following indefinite integral: \begin{equation} \int\frac{x^3+1}{x+1}dx. ...
7
votes
2answers
103 views

Calculate the integral

I have been trying to solve this integral which arises in a problem from the book Fundamentals of Differential Equations Nagle, Saff and Snider (8th edition - site 193-17). $$\int\frac{e^x ...
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0answers
48 views

About calculating limits of integrals (Part 1)

Say $a, x$ and $y$ are three real number constants and $t$ is a real variable. Now define the complex number, $z = -y + i(a+x-t)$ and consider an integral of the form $\int_{t= f(x,y)}^{t = g(x,y)} ...
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1answer
73 views

Find the area of the region above the $x$-axis bounded by the line $y=4x$ and the curve $y=x^3$?

Find the area of the region above the $x$-axis bounded by the line $y=4x$ and the curve $y=x^3$ Attempt: intersect when:- $x^3 - 4x = 0$ $x ( x² - 4 ) = 0 $ $x = 0 , x = \pm 2$ Area is given ...
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0answers
50 views

Moment generating function, Variance and Expected Value

Let $M_X (t)$ be a moment generating function. These are the things I know: $Mean=E[X] =M'_X(t)$ at $t=0$ (likewise the second expected value is the second derivative evaluated at $t=0$) $V[X] = ...
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2answers
45 views

Finding the function given its derivative and some conditions.

Let $f:[0,\frac{\pi}{2}]\to \mathbb R$ be continuous and satisfy $f'(x)=\frac1{1+\cos(x)}$ for all $x\in(0,\frac\pi2)$. If $f(0)=3$ then $f(\frac\pi2)$ has the value equal to: One would simply do ...
2
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1answer
56 views

Evaluating $\int^{\frac\pi2}_{\frac\pi4}(2\csc(x))^{17}dx$

I saw this question in JEE Advanced. But in that we had to simplify it to $$\int^{\log(1+\sqrt2)}_{0}2(e^u+e^{-u})^{16}du$$ But I pose the following question as to evaluate this integral in closed ...
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1answer
15 views

Symmetries of a function imply certain properties of its Fourier coefficients

Exercise from Fourier Analysis: An Introduction by Stein and Shakarchi: Let $f$ be a $2\pi$-periodic Riemann integrable function defined on $\Bbb R$ with $f(θ + π) = f(θ)$ for all $θ \in \Bbb R$. ...
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2answers
198 views

$f'\in \mathcal R([0,1])$ , then $\lim_{n \to \infty} \sum_{k=1}^n f\Big(\dfrac kn \Big) - n \int_{0 }^1 f(x)dx=\dfrac{f(1)-f(0)}2$?

If $f:[0,1]\to \mathbb R$ is a differentiable function with continuous derivative then I can show that $$ \lim_{n \to \infty} \left[ \sum_{k=1}^n f\!\left(\dfrac kn \right) - n \int_0^1 f(x)\,dx ...
3
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1answer
43 views

How do you find the Fourier transform of a function?

I will illustrate this with a simple example: Consider the exponential decay function $$f(t)=\begin{cases} 0 & \ t\lt 0 , \\ A e^{-\lambda t} & \ t\ge 0 \end{cases}$$ Where $\lambda ...
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2answers
130 views

What is the relation between Fourier's Inversion theorem and the Dirac-Delta function?

This is a direct quote from page 472 of this book: From Fourier's Inversion theorem $$f(t)= \int_{-\infty}^\infty f(u) \, \mathrm{d}{u} \left( \frac{1}{2\pi}\int_{-\infty}^\infty ...
2
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2answers
342 views

Definite integration - involving greatest integer function

Question: Find the value of the integral: $$\int^{3\pi\over 2}_{\pi \over 2}[2\sin x] dx$$ Where $[y]$ represents the greatest integer less than or equal to $y$. I know that I will have to ...
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0answers
85 views

Examples of integrals solved using hyperbolic functions.

I've read in some questions here that various types of integrals usually solved by involving $\tan$ and $\sec$ into the mix can sometimes be solved in an easier manner using hyperbolic functions, as ...
2
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5answers
142 views

Evaluate $\int_{0}^{\frac{1}{2}}\ x\cos(\pi x)\,\mathrm{d}x$

Evaluate$$\displaystyle\int_{0}^{\frac{1}{2}}\ x\cos(\pi x)\,\mathrm{d}x$$ My $u = x$ and my $du = dx$ $dv = \cos(\pi x)\, dx$ $v=\sin(\pi x)$ The answer book however has $v=\frac{1}{\pi}\sin(\pi ...
2
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2answers
33 views

Proof for formula $\int e^{g(x)}[f'(x) + g'(x)f(x)] dx = f(x) e^{g(x)}$

I recently saw someone using this formula here on one of the questions since I can't comment , can someone please give me the proof of this equation and type of problems where it can be used ?
1
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1answer
43 views

Derive the expectation of the r-th inverse moment

$X$ is nonnegative, $\phi(t) = E[e^{-tX}]$ is finite for $t \geq 0 $. Show that for any $r > 0$, $E(\frac{1}{X^r}) = \frac{1}{\Gamma(r)} \int_{0}^{\infty} t^{r-1} \phi(t) dt.$ Thanks in advance. ...
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1answer
83 views

How to compute this contour integral with a modulus sign in the integrand,

Evaluate the integral $$∫_{∣z∣=ρ} \frac {1}{|z−a|^{2}}|dz|$$ where ρ is a positive number, a is a complex number, and |a|<ρ. I welcome any hints on how to get started on this problem. The ...
2
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2answers
119 views

$\int_{-1}^{+1}\sin(\sqrt{1-x^2})\cos(x)\,dx$ =?

I am trying to find the value of the definite integral: $$\int_{-1}^{+1}\sin\left(\sqrt{1-x^2}\right)\cos(x)\,dx$$ The answer from WolframAlpha is $1.20949$ but I can't solve it analytically.
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0answers
53 views

express $\int(x^2ln(1+x))$ as a power series

$\int(x^2ln(1+x))\,dx$ My work: $x^2ln(x+1)$ = $x^2\int\frac1{1+x}\,dx$ = $x^2\sum_{k=0}^\infty(-1)^kx^k$ = $$\sum_{k=0}^\infty (-1)^kx^{(k+2)}$$ Integrating gives: $$C+\sum_{k=0}^\infty (-1)^k ...
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0answers
28 views

Probability Density Function Integration Question

So this question is almost exactly the same as a few others on here but the vital piece of information I need is missing. Consider an RV X with a pdf: $f(x) = xe^{−x^2/2} $ if $ x\gt 0$ $f(x) = ...
0
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1answer
34 views

Uniform Limit vs. Integral Mean

Denote uniform norm: $$\|f\|:=|f(t)|_{t\in\mathbb{R}}:=\sup_{t\in\mathbb{R}}|f(t)|$$ Consider a sequence: $$f_n\in\mathcal{C}(\mathbb{R}):\quad\|f-f_n\|\stackrel{n\to\infty}{\to}0$$ Then does it ...
2
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2answers
66 views

Proof of generalization of Divergence theorem

I looked around the Web for a proof of what I have heard called "the Divergence theorem". However, it seems this name implies three-dimensionality, as all I could find was a series of proofs like this ...
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1answer
112 views

Closed form for $\int_0^R \frac{dx}{\sqrt{\ln(1+x)}}$, R>0

I stumbled on an interesting integral doing some physics exercise which did not require its closed form (if it has any). It has, however, sparked my interest and I tried my best to find it, but I ...
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0answers
51 views

No “Conditional Convergence” in Lebesgue Integration

In my measure theory course, I've heard the professor say numerous times that "there is no conditional convergence in Lebesgue integration, it is always absolute." As an example, he shared ...
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1answer
61 views

Why is a definite integral used in solving this differential equation?

The following equation is given in my notes $$\frac{\partial u}{\partial t} = D\frac{\partial^2 u}{\partial x^2} \quad\textrm{on domain} \; |x| < \infty, \, t>0$$ The author starts by doing the ...
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3answers
46 views

Integration - finding range of function [closed]

Question: Let $f:\big[\frac{1}{2},1\big] \to \text R$ be a positive, non-constant and differentiable function such that $f'(x)<2f(x)$ and $f\big(\frac{1}{2}\big) = 1$. Then find the interval in ...
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3answers
45 views

Lebesgue integral/measure issue

So I have an exercise that seems trivial to me, although, I could have done the proof completely wrong; I'm worried my negation is wrong. Here is the statement: Given $\epsilon > 0$, show that ...
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1answer
55 views

Solving a non-linear Partial differential equation $px^5-4q^3x^2+6x^2z-2=0$

I have to find out the complete integral of : $px^5-4q^3x^2+6x^2z-2=0$ My attempt:Let $f(x,y,z,p,q)=px^5-4q^3x^2+6x^2z-2$ So, $f_p=x^5,f_q=-12q^2x^2,f_x=5px^4-8q^3x+12xz,f_y=0,f_z=6x^2$ Applying ...
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0answers
39 views

Solving integrated squared density derivative with integration by parts

I'm trying to understand why $\int f^{(s)}(x)^2 dx = (-1)^s \int f^{(2s)}(x)f(x) dx$ is valid (under sufficient smoothness assumptions on $f$ where $f$ is the density of some sample). It should be ...
4
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1answer
69 views

The distance to smooth boundary, raised to a power between $-1$ and $0$, is integrable

Let $\Omega$ be a bounded domain of $\mathbb{R}^N$ with smooth boundary. Show that ${\rm dist}(x,\partial \Omega)^{-\vartheta} \in L^1(\Omega)$ if $\vartheta \in (0,1)$. I can see why if $\Omega$ is ...
1
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1answer
47 views

Find the integration of $\int_{-\pi}^{\pi} (10cos10t+20cos20t)^2dt$

Could you help me to find solution of $$\int_{-\pi}^{\pi} (10\cos10t+20\cos20t)^2dt$$ I have solution. But I did not understand why the term $400\cos10tcos20t$ disappear. Thank you
0
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1answer
34 views

The integral of $\frac{1}{\tau_1-\tau_2}(e^{-t/\tau_1}-e^{-t/\tau_2})$

If we have $$ \frac{dx}{dt}=\frac{1}{\tau_1-\tau_2}(e^{-t/\tau_1}-e^{-t/\tau_2}), $$ what is $x$, and what are the steps by which one comes to that solution?
2
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1answer
29 views

Compute the integral for $C$ given by $e^{i\theta}$, $\theta \in [0, 2\pi]$

Here is the problem : $$\int_{\mathcal{C}} \frac{z^9+e^{iz}+(7654)^{\sin(z)}z}{z-1} \,\mathrm{d}z,$$ for $C$ given by $C(\theta) = e^{i\theta}/2, \theta \in [0, 2\pi]$ I think that I can use the ...
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0answers
31 views

Lebesgue Integral of Characteristic function.

Let $A_n = [1-n,n-1]$. Define $$\chi_{\mathbb{R} \setminus A_n} := \begin{cases} 1 \qquad x \in \mathbb{R} \setminus A_n \\ 0 \qquad \text{ else } \end{cases}$$ Is it true that $$ \lim_{n \to ...
0
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1answer
29 views

Compute $\int_{\gamma}\frac{dz}{z\sin(z)}$

Compute $\int_{\gamma}\frac{dz}{z\sin(z)}$ where $\gamma:[0,2\pi]\to \Bbb{C}$ is given by $\gamma(t)=e^{it}$ Im having a problem with this integral since, $\frac{1}{\sin(z)}$ is not holomorphic in ...