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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1answer
27 views

If $f$ is continuously differentiable periodic, then $n\int_{0}^{1} f(x) \sin (2\pi nx) \mathrm dx \to 0 $

If $f: \mathbb R \to \mathbb R$ is continuously differentiable periodic function of period $1$, then $$n\int_{0}^{1} f(x) \sin(2\pi nx)\mathrm dx \to 0 $$ as $ n\to\infty$.
3
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2answers
107 views

Modified Leibnitz integral: $\lim\limits_{a \to\infty}\frac1a\int _0^\infty\frac{(x^2+ax+1)\arctan(\frac{1}{x})}{1+x^4}dx=?$

$\lim\limits_{a \to \infty} \frac{1}{a} \int _0^\infty\frac{(x^2+ax+1)\arctan(\frac{1}{x})}{1+x^4}dx $ ,where $a$ is a parameter. ATTEMPT:- Let $I(a)=\frac{1}{a} \int _0^\infty ...
1
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2answers
35 views

Problem of a solid of revolution. When I need to use the $2\pi$ in the shell method. Exactly why is used that $2\pi$

The semicircular region limited by $x=\sqrt{4-y^2}$ and the Y-axis and revolves around the line $x =-1$. generated and calculate its volume. First: the function; I can't make the graph. So, I´m not ...
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2answers
37 views

What is the name of this kind of integral called?

$$f_n(x) = \frac{1}{n}\int_x^{x+n} f(t) dt$$ What is this kind of integral called? Thanks!
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1answer
34 views

Problem in solving an Integral.

I'm solving a solid of revolution problem and I'm stuck at this point. The u-subtitution doesn´t work, I don´t know what method use. $\pi\int(\frac{4x-1}{8x^4})^2$
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1answer
39 views

About the “Bounded Convergence Theorem”

The ``Bounded Convergence Theorem" states that "If a sequence $\{f_n\}$ of measurable functions is uniformly bounded and if $f_n \rightarrow f$ in measure then $lim_{n \rightarrow \infty } \int f_n ...
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1answer
80 views

Why can't Fubini's/Tonelli's theorem for non-negative functions extend to general functions?

Challenging-conventional-wisdom question based on an answer to my previous question. If $X \in L^1 (\Omega, \mathscr{F}, \mathbb{P})$ has pdf $f_X$, $Y \in L^1 (\Omega, \mathscr{F}, \mathbb{P})$ has ...
0
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1answer
82 views

Questions on Doob's Optional Stopping Theorem (a) and (b)

From Williams' Probability w/ Martingales: What is $X_T$ in red box above? I am fairly certain this was not defined previously in the book. There was this though: I have a feeling $X_T = ...
4
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1answer
73 views

Let $a_f=\text{ arg} \min_{a} \int \left|f(x)-a\right| dx$ and $a_g= \text{ arg} \min_{a} \int \left|g(x)-a\right| dx$, is $a_f \le a_g$?

Let $ f(x) \le g(x) $ and assume that $g(x),f(x) \in L^1$ let \begin{align} a_f= \text{ arg} \min_{a } \int_A \left|f(x)-a\right| dx\\ a_g=\text{ arg} \min_{a } \int_A \left|g(x)-a\right| dx ...
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0answers
32 views

Finding upper bound for sum

My question: Show that an upper bound for the sum $$\sum_1^{10}\frac{r+1}{r^2+2r+2}$$ is equal to $\ln 61$ My attempt: Upper bound is equal to $$\int_1^{11}\frac{r+1}{r^2+2r+2} dr$$ ...
1
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1answer
52 views

Log-linearizing $Y_t=\int_0^1 F(X_{it}) di$

I want to prove that log-linearizing the expression $Y_t=\int_0^1 F(X_{it}) di$ yields: $$Yy_t \approx F'(X)X\int_0^1 x_{it} di$$ Where: $\{X_{it}\}_{i \in (0,1)}$ is a continuum of strictly ...
1
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2answers
48 views

What does it mean by the Integration notation? [duplicate]

$\int_a^b f(x) dx = 1 -------------------(*)$ I know that this means: (1) Integral of $f(x)$ within $a$ and $b$ is $1$. or, (2) The area under the curve represented by the function $f(x)$ ...
2
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2answers
28 views

Show that $\int_0^1 f^2(x)dx\geq 3$, if $f:[0,1]\to \mathbb{R}$ is an integrable function s.t. $\int_0^1 f(x)dx=\int_0^1 xf(x)dx=1$

So far I've done this, but I don't know if it will help. Let $\int_0^xf(t)dt=F(x)$. Now, $1=\int xf(x)=F(x)x|_0^1-\int_0^1F(x)dx=1-\int_0^1 F(x)dx$. Then, $\int_0^1 F(x)dx=0$. But I don't know how ...
2
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3answers
80 views

Can someone explain the integration of $\sqrt{v²+\tfrac14}$ to me?

I am currently trying to integrate this root: $$\sqrt{v^2+\frac{1}{4}}$$ According to several integration calculators on the web it is: $$\frac{\operatorname{arsinh}(2v)}{8} ...
1
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1answer
30 views

Breaking up integral representations by convergence

A known integral takes the form of $$\zeta(3)=\frac{1}{2}\int_{0}^{\infty} \frac{t^2}{e^t-1}dt$$ Through Wolfram part of the integral converges to $$\int_{0}^{\infty} \frac{t}{e^t-1}dt = ...
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0answers
50 views

A set of integers related to $\pi$

Let $\Psi$ be the maps \begin{array}{lrcl} \Psi : & {\mathbb N}^{(\mathbb N)} & \longrightarrow & \mathbb R^+\cup\{+\infty\} \\ & A=(a_i)_{i\in \{0,\ldots,n-1\}} & \longmapsto ...
2
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3answers
332 views

Integration with base 10

I have to calculate the following integral: $$ \int_{1}^{2}10^{x/2}dx $$ I simplified it as follows: $$ ...
2
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1answer
27 views

Finding the double integral over a specified region.

Let $D$ be the region in the xy-plane that is bounded by the coordinate axes and the line $x+y =1$ , we need to find : $\displaystyle\iint (x-y)e^{ x^{2} + y^{2}}dydx$ over $D$. I am trying the ...
2
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1answer
50 views

To prove $\int_{a}^{\theta} f(t)dt = \theta f(\theta)$.

Given a real continuous function f on [a,b] such that $\int_{a}^{b} f(t)dt = 0$ where a>0. Prove that $\exists$ $\theta$ where $a<\theta<b$ such that $\int_{a}^{\theta} f(t)dt = \theta ...
2
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0answers
401 views

Integral inequality - lower bound on $L^1$ norm.

I was wondering if one can make an estimate of form: Assume $f\in C^\infty(\overline{\Omega})$ where $\Omega$ is a bounded domain in $\mathbb{}R^d$. Is there a constant $C>0$ independent of $f$ ...
4
votes
3answers
67 views

Integrate $\int r^n \sin r \,dr$

How to compute $\int r^n \sin r\, dr, n\in \Bbb Z$? In fact, I really need 10 reputation points to ask a complex question.
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2answers
75 views

Minimizing $\int_{0}^{1} (1+x^2)f^2(x)dx$

What is $$\min_{f\in D} \int_{0}^{1} (1+x^2)f^2(x)\mathrm dx,$$ where $D$ is the collection of all continuous real functions from $[0,1]$ such that $\int_{0}^{1} f(x)$ = 1. My attempt Note that ...
6
votes
4answers
310 views

Use integration by parts to find the integral $\int\frac{\sqrt {4x^2-9}}{x^2}dx$

$$\int\frac{\sqrt {4x^2-9}}{x^2}dx$$ I tried to solve this using integration by parts, but I come up with something that is much more difficult to solve. How can this be solved?
1
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3answers
51 views

How to evaluate the following integral without using partial fraction?

$$\int \frac{dx}{(x^4-1)^2} $$ One way would be applying partial fractions. But Its very tedious. Can you suggest any other way ?
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1answer
75 views

Extension of Cauchy Integral Formula

I'm now taking a course in complex analysis and in wikipedia it was said that Cauchy Integral formula is true also for a function which is "holomorphic in the open region enclosed by the path and ...
2
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1answer
21 views

Finding the value of integers of a logarithmic function.

Expressing $\frac{2x^2-9x-6}{x(x^2-x-6)}$ in partial fraction would give you: $\frac{16}{5(x+2)}$-$\frac{1}{5(x-3)}$-$\frac{1}{x}$ Given that $\int_{4}^6\frac{2x^2-9x-6}{x(x^2-x-6)}dx=ln\frac{m}{n},$ ...
2
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1answer
50 views

Does $\int \frac{exp( -b\sqrt{a+x})}{\sqrt{x}} dx$ have a solution?

Is there a solution for the following integral: $$ \int_0^{\infty} \frac{\exp( -b\sqrt{a+x})}{\sqrt{x}} dx $$ where a and b are constants. If it is not, what is the best approximation? Especially ...
2
votes
3answers
126 views

Evaluating an infinite sum involving possibly hypergeometric terms

I was considering the following infinite sum $$ A(n) = \sum_{k=2}^{\infty}\left[\frac{(-1)^{k+n-1}}{k^n}(0k -1)(k-1)(2k-1)...((n-1)k-1) \right] $$ Some cases: $$ A(1) = ...
1
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3answers
88 views

Finding derivative with integral

Came to this problem on my test study guide. I thought for finding the derivative you just took the antiderivative, which would be -cos(theta). How is it sin(x)? Or did I come across another error? ...
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0answers
33 views

Closed-form expression for this integral?

A popular Internet image that has a double integral is: However, even WolframAlpha cannot evaluate the integral: $\int \int |x^{y^x} y^{x^y}| e^{xy} dy\;dx$. Is there a closed-form expression for ...
2
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1answer
89 views

Richardson's extrapolation of composite trapezoidal rule

I have applied Richardson's formula to the composite trapezoidal rule, $I_h(f)=\frac{h}{2}(f(a)+\sum_{k=1}^{n-1}f(a+kh)+f(b))$, in an attempt to better approximate the integral $I(f)=\int_0^1 ...
1
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1answer
50 views

Finding $\overrightarrow v(t)$ and $\overrightarrow r(t)$ when given $\overrightarrow a(t)$

Suppose an object moves so that its acceleration is given by $\overrightarrow a(t) = \langle 0, -4 \cos(t), -3 \sin(t) \rangle$ with $\overrightarrow v(0)= \langle 0,0,3 \rangle$ and ...
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0answers
42 views

Riemann integrals: invariant under isometry?

Let $T:\mathbb{R}^n\to\mathbb{R}^n$ be an isometry and let $f:D\to\mathbb{R}$ be a Riemann integrable function on $D$, i.e. such that the limit $$\int_D f(x_1,\ldots,x_n)\,d x_1\ldots ...
3
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4answers
192 views

How to integrate this fraction: $\int\frac{1}{1-2x^2}dx$?

I'm not sure how to integrate this: $$\int\frac{1}{1-2x^2}dx$$ I think it has to be this: $$ -2\cdot \arctan(x)$$ Or this: $$\arctan(\sqrt{-2x^2})$$
2
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0answers
17 views

Special name and the general solution method for the given integral

I wonder whether the integral $$ I = \int \frac{d^3p}{(2\pi)^3}\frac{e^{i\vec{p} \cdot \vec{x} }}{\vec{p}^2+m^2}, $$ has a special name or not and besides I am also interested in the general method ...
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1answer
28 views

Show $f'(\beta) \int\limits_\beta^\alpha 1 dt \leq \int\limits_\beta^\alpha f'(t) dt \leq f'(\alpha) \int\limits_\beta^\alpha 1 dt$

Assuming $f(x)$ is a function of single variable, and $f'(x)$ is monotonically increasing Then claim: $f'(\beta) \int\limits_\beta^\alpha 1 dt \leq \int\limits_\beta^\alpha f'(t) dt \leq f'(\alpha) ...
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2answers
61 views

Find the area of the region inside the inner loop of the​ limaçon

Find the area of the region inside the inner loop of the​ limaçon $r=7+14\cos(\theta))$ So doing this problem, i got B the integral from 0 to 2pi (1/2)($7+14\cos(\theta))$^2. and the Area as ...
0
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1answer
19 views

Finding the double integral when the boundaries for y is not specified

A question in my Calculus book states: "Find the volume of the solid in the first octant bounded by the cylinder $z=9-y^2$ and the plane $x=2$" When this answer was being covered in class, it was ...
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2answers
150 views

Give 5 proofs for $\int_0^{2 \pi} \ln(\frac{25}{16} - \sin(x)^2) dx = 0$.

When you ask my mentor : Am I any good at integrals ? You usually get an answer like this : Give 5 proofs for $\int_0^{2 \pi} \ln( \frac{25}{16} - \sin(x)^2) dx = 0$. I was able to show it with ...
4
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2answers
126 views

Evaluation of $\int \sqrt{1+\cot x}dx$?

What is $$\int \sqrt{1+\cot x}dx$$ My friends and I tried using all possible trigonometric formula. We couldn't find a way to solve it Please help me solve it.
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0answers
46 views

Convolution of exponential and rect functions

I have a convolution question in my signals and systems problem set that is puzzling me: $ f(t) = e^{-t/2T} u(t) $ and $ g(t) = rect(t/2T) $ find the convolution $f \ast g$ and I am assuming ...
2
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1answer
63 views

Completely missing the idea of this solution (finding an arbitrary function in integrand)

I'm hoping that someone can explain this (partial) solution to me. In my textbook (Haberman PDEs book, q. 10.2.1), we're asked to find (complex) $c(\omega)$ so that the following are equivalent (with ...
2
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2answers
65 views

Approximation for integral of Matrix Exponential

I am trying to implement some algorithm in matlab. To this end, I need to discretise a system of differential equation as $\dot x = A x + B u$. Starting with initial condition $x_0$ the system after ...
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2answers
125 views

Prove that $ \int_0^1 x^2 \psi(x) \, dx = \ln\left(\frac{A^2}{\sqrt{2\pi}} \right) $

Basically what the title says: Prove that $ \displaystyle \int_0^1 x^2 \psi(x) \, dx = \ln\left(\dfrac{A^2}{\sqrt{2\pi}} \right). $ where $A\approx 1.2824$ denotes the Glaisher–Kinkelin ...
1
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1answer
42 views

Help with integration involving exponential

I am trying to solve an equation in the book Digital Image Processing, but I am stuck in the steps in between the formula and solution. Here's the equation, the last line is the solution. Sorry it's ...
1
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1answer
45 views

Recursive formulas and integration [duplicate]

Using integration by parts find a recursive formula of $\int cos^n(x) dx$ and use it to find $\int cos^5 x dx$ I have no idea how to do this and my knowledge does include integration by parts etc. I ...
0
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0answers
17 views

Domain in polar coordinates with a square and a discus

I was doing some studying in Steward's Calculus when I came onto this problem. I am asked to integrate a certain function $f(x,y)$ in this domain. I know how to do it when the inner boundary is a ...
3
votes
5answers
178 views

Integral $\sqrt{1+\frac{1}{4x}}$

$$\mathbf\int\sqrt{1+\frac{1}{4x}} \, dx$$ This integral came up while doing an arc length problem and out of curiosity I typed it into my TI 89 and got this output ...
2
votes
1answer
48 views

Can I use my approach to show that the integral of f(x)cos(nx), over one period, goes to zero?

Here's my work, without measure theory (Riemann-Lebesgue lemma) and without step function approximations: Let f(x) be continuous on the closed interval $[0,2\pi]$, and $x$ a real variable. Then we ...
0
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0answers
32 views

Calculation of surface integral without parameters

Yesterday ago we learnt about surface integrals, and I already calculated some with parameters. One of these exercises was this one: $x= v\cos(u)$ $y= \sin(u)$ $z= v$ while $B' = {(u,v): 0 \le ...