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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0answers
9 views

Estimation of a certain Integral

I estimated (w.r.t. $\epsilon$) the expression $|\displaystyle\int_{-1}^{x_0-\epsilon} (1-x)^{n-p}(1+x)^{p}+\displaystyle\int_{x_0+\epsilon}^{1} (1-x)^{n-p}(1+x)^{p} dx |$ ...
0
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0answers
8 views

Prove that an integral from q

Let the series $\left ( I_n \right )_{n\geq 0}$ , $I_n=\int_{0}^{1}\frac{\left ( x^2+x+1 \right )^n - 1}{x^2+1}dx$ c)Prove that $I_{4_n+1} \in \mathbb{Q}$ .
3
votes
6answers
80 views

Error in my proof?

What is wrong in this proof. It seems correct to me but still doesn't make proper sense. $=\sqrt{...\sqrt{\sqrt{\sqrt{5}}}}$ $=5^{1/\infty}$ $=5^0$ $=1$ EDIT So does this mean that ...
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1answer
20 views

The Gherkin (an egg shaped building) - equation for the curve in order to calculate the surface area of revolution

I am trying to calculate the surface area of revolution for The Gherkin, an egg-shaped building in London, UK. Not sure about how to obtain the equation of the curve but I have the data points that ...
0
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1answer
11 views

Reasoning behind method of steepest descent

I am considering the method of steepest descent from my notes. I have written that $$\int_a^b dx e^{g(x)} \sim e^{g(x_0)} \int_{\infty}^{\infty}dx \exp \left[-\frac{1}{2}(x-x_0)^2|g^"(x_0)|\right] ...
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3answers
35 views

Arc-length of an Archimedean Spiral

I want to calculate the arc-length of the archimeadean spiral given by the equation: $\vec{x}(t)=\begin{pmatrix} e^{-\alpha t} \cos t \\ e^{-\alpha t} \sin t\end{pmatrix}$ $\alpha >0$ and $t \in ...
1
vote
1answer
43 views

Find this integral $I(x)=\int_{0}^{+\infty}\frac{1}{y}e^{-y-\frac{x}{y}}dy$

Find this integral $$I(x)=\int_{0}^{+\infty}\dfrac{1}{y}e^{-y-\dfrac{x}{y}}dy$$ I think $$I'(x)=-\int_{0}^{+\infty}\dfrac{e^{-y-\frac{x}{y}}}{y^2}dy$$ Now I have no idea of how to continue
1
vote
1answer
24 views

Evans PDE: Chapter 5, Problem 9 - Clarification

I've been trying to work out the solution to Question 9 in Chapter 5 of Evans, and I'm having some difficulties. I've been looking at the solution posted here: question 9 - chap 5 evans PDE And I ...
0
votes
4answers
75 views

Area of the curve sin(cos(x))

Find the area of the region enclosed by the curves $y = \sin (\cos(x))$, $y = 0$ ,$x = π / 2$, and $x = −π / 2$. I am not able to integrate the function. How do I find this area?
5
votes
2answers
33 views

Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$.

(Jones, p. 246) Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$. This seems pretty easy to prove in the following way: Let $g_j$ be a ...
2
votes
1answer
34 views

Riemann Integral on $\mathbb{R}^2$

I have the following question. Find a function $f(x,y)$ that is integrable on rectangle $[0,1] \times [0,1]$, such that $g(y) = f(\frac{1}{2}, y)$ is not integrable for $y \in [0,1]$, or prove that ...
1
vote
1answer
16 views

Integration about x and y axes to find area

I have a problem statement that requires me to find area between the curves about x axis and about y axis. But my answers are not matching. Please find below my worked out solution - The ...
2
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1answer
11 views

parallelepiped change of variables

I can't figure out how to start this problem. Use a triple Integral to find the mass of a parallelepiped generated by the vectors $$<6,1,2>,\ <3,3,9>,\ {\rm and}\ <2,7,3>.$$ We are ...
5
votes
5answers
105 views

Proving $\int_{0}^{1}\sqrt{\frac{1-x}{1+x}}dx=\frac{π}{2}-1$

Proving $$\int_0^1 \sqrt{\frac{1-x}{1+x}} \, dx= \frac{π}{2}-1$$ My attempt is: I assumed the $1-x=u$ $du =-dx$ $$\int_0^1 \sqrt{\frac{u}{2+u}}\,(-du)$$ here I stoped and I couldn't how to complete ...
1
vote
1answer
57 views

Definite integral involving 2015

Evaluate $$\displaystyle\int_{2}^{2014} \frac{\log \left( 2015 - x\right )}{\log \left( 2015 - x\right ) + \log \left( x - 1\right )} \mathrm{d}x$$ I got the solution using software, and it is a ...
0
votes
0answers
43 views

how to solve this integral involving any square root

how to solve the integral $\int\sqrt{\alpha+\beta e^{\gamma t}}dt$ i got this integral from the problem Given that the velocity $v$ of a body $t$ segonds after passing a point $O$ is found by ...
4
votes
2answers
69 views

computing an integration with a floor function

I am trying to compute $$\int_0^1 \left(\frac{1}{x} - \biggl\lfloor \frac{1}{x}\biggr\rfloor\right) dx$$ with no success. Any hints?
8
votes
1answer
90 views

Derivative of $\int_0^1 e^{\sqrt{x^2+t^2}}\,\mathrm{d}x$ at $t = 0$

Let the real-valued function $\phi:\mathbb{R}\to\mathbb{R}$ be defined by $$\phi(t)=\int_0^1e^{\sqrt{x^2+t^2}}\,\mathrm{d}x,$$ it can then be shown that $\phi$ is continuous and differentiable. I ...
0
votes
3answers
78 views

Integrating $f(x) = 1/x$ from $x=a$ to $x=\infty$

Can the integration of $f(x)=1/x$ from $x=a > 0 $ to $x=\infty$ ever be finite? That is, can $\int_{x=a}^{\infty} 1/x$ be finite?
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votes
5answers
127 views

Is it possible to calculate for example $\int_{0}^{1} x \mathrm{d}2x$

My question is just for fun, but I want also to verify if I understand something in variation calculus... I want to know if it is possible to calculate this : $$ \int_{0}^{1} x \mathrm{d}2x $$ A ...
1
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2answers
23 views

Difference between Path, Curve, Graph and Trace

I am having difficulties in understanding the differences between these concepts. We have a new lecturer who loves writing down things in dense mathematical notation (I don't think that's bad but I am ...
4
votes
5answers
75 views

Something wrong at $\int \frac{x^2}{x^2+2x+1}dx$

I have to calculate $$\int \frac{x^2}{x^2+2x+1}dx$$ and I obtain: $$\int \frac{x^2}{x^2+2x+1}dx=\frac{-x^2}{x+1}+2\left(x-\log\left(x+1\right)\right)$$ but I verify on wolfram and this is equal with: ...
2
votes
1answer
58 views

Arithmetic mean of $L^2$ function is $L^2$

I have found the following problem, to which I do not find the solution: Consider $f(x), x > 0$ a function such as $$ \int_0^\infty f^2(x) dx < \infty $$ and let $g(x) = \frac 1x \int_0^x ...
0
votes
0answers
24 views

Confusion about Flow Integral

I am asked to calculate the flow integral $$\int{\vec{F}\cdot\hat{T}ds}$$ of $\vec{F}=<2x,-3y>$ along the fourth quadrant path from (5,-3) to (8,0) along the curve $x^2-10x+y^2+16=0$. So I ...
0
votes
0answers
22 views

Area between curves (Calc) [on hold]

How do you know whether to solve with respect to y or respect to x? I know how to do the rest of the problem, I never know which one to do though.
0
votes
1answer
34 views

How to integrate this using tan(x/2) substitution?

How do I integrate cos(x)/(sqrt(5)+cos(x)) ? I have been advised to use t = tan(x/2) substitution but ended up with a polynomial of degree 4 over one of degree 6 to integrate, which did not have an ...
0
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0answers
28 views

taking integral of odd functional form

I'm currently working on a paper and have come across the following integral: $$\int \frac{f'(x)}{(x - f(x))} dx$$ I'm not familiar with any typical way of evaluating the integral and plugging it ...
0
votes
2answers
56 views

Where is the mistake in this integral? [duplicate]

Consider the following simple integral. \begin{align*} \int\dfrac{1}{2x}\,dx. \end{align*} Now, I would usually work as follows: \begin{align*} ...
0
votes
1answer
87 views

Evaluate $\int\sin(x^2)\mathrm{d}x$

Can you evaluate $$\int\sin(x^2)\mathrm{d}x \quad ?$$ I have tried substituting $p=x^2$ as well as integrating by parts, but then I came across an answer here which says that there is no way of ...
1
vote
1answer
40 views

How do I integrate this expression?

Need to find the following: $$\int \frac{\sqrt{x^2 - 4}}{x^3} dx$$ So far, after using integration by parts, I've arrived at $\frac{-\sqrt{x^2-4}}{2x^2} + \frac12 \int \frac{1}{x\sqrt{x^2-4}} dx$. ...
1
vote
1answer
23 views

Integrability of dirichlet function in $\mathbb{R}^3$

Let $d: [0,1] \rightarrow \mathbb{R}$ be the Dirichlet function as follows: $$d(x) = \begin{cases} 1, & x \in \mathbb{Q} \\ 0, & x \in \mathbb{R} \backslash \mathbb{Q} ...
0
votes
0answers
15 views

About minimal curvature of splines

I am given a the following problem set: Let $s$ be a natural cubic spline that interpolates a function $f \in \mathcal{C}^2 ([a,b])$ at points $a = x_0 < x_1 < \ldots < x_n =b$ with ...
0
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1answer
30 views

The period of a non-linear pendulum

The period of a non-linear pendulum is $T = \sqrt{2} \cdot \int_{-\theta_0}^{\theta_0} \frac{d{\theta}}{\sqrt{\cos(\theta) - \cos(\theta_0)}}$. My problem: what will happened with the period $T$, ...
0
votes
1answer
19 views

How to use Cauchy's integral formula with more than one pole?

$\int\limits_{\gamma} \frac{z^2}{z(z-2)}$ $\gamma(\theta) = 3e^{i\theta}$, $0 \leq \theta \leq 2\pi$ Cauchy's integral formula is given by: $$\int\limits_{\gamma} \frac{f(z)}{(z-a)^{n+1}} = ...
0
votes
0answers
23 views

Double Integral with integrand similar to bivariate normal density

I got a double integral like the following, $$\int_{0}^{\infty} \int_{y}^{\infty} xe^{-\frac{(x-by-c)^2}{2a}}ye^{-\frac{(y-e)^2}{2d}}dxdy,$$ where $a$,$b$,$c$,$d$,$e$ are viewed as some other ...
0
votes
0answers
18 views

Period of a non-linear pendulum

I have proved that the period of a non-linear pendulum is $T(\theta_0) = \sqrt{2} \cdot \int_{-\theta_0}^{\theta_0} \dfrac{d{\theta}}{\sqrt{\cos(\theta) - \cos(\theta_0)}}$. I need to show, that ...
-1
votes
0answers
28 views

What does u mean in density function [on hold]

I have such hometask - We have a density function: $$ f_X(x) = \frac{1}{2} u(x)\,\exp\left(-\frac{3x}{2}\right) $$ u is a step function So I need to find expected value of $X^3$. How to solve this?
4
votes
2answers
87 views

Evaluate $\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$

Evaluate $$\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$$ For this integral, I have tried using integration by parts and then ...
1
vote
2answers
38 views

Substituting an even function for an odd function in an integral

In a book I am reading it says while evaluating an integral it says you cannot substitute an odd function for an even function of another variable ( for example substituting sinx with cosu and ...
0
votes
1answer
23 views

How to determine the area of the paraboloid enclosed by the cone?

Is it possible to determine the exact area of the paraboloid that falls inside the cone? I've been trying for days without success...
-1
votes
0answers
16 views

Integral of the product of three low-order Spherical Harmonics

Given two functions $f(x)$ and $g(x)$ approximated by Spherical Harmonics with the coefficients $c_i^f$ and $c_i^g$, the integral of the product of the two (approximated) functions can be calculated ...
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votes
0answers
30 views

Need help understanding the derivation shown

I have been trying to figure out how to go from eq. (15) to (17) using (15a),(15b) and (16). The rhs of (15) and its transformation to the $-24/T^4$ term in (17) is straightforward so can be ignored. ...
0
votes
1answer
26 views

Divergence of $\iint \text e^{-(\tau_1-\tau_2)}\,\theta(\tau_1-\tau_2)\,\text d ^2\tau$

Does this integral ($\alpha>0$) $$ I=\int_{-\infty}^\infty\text d \tau_1 \int_{-\infty}^\infty\text d \tau_2 \; \text e^{-\alpha(\tau_1-\tau_2)}\theta(\tau_1-\tau_2) $$ diverge? Here $\theta$ is ...
3
votes
1answer
101 views

show that integral $\int_{0}^{\infty}y\,dx=\frac{\pi^2}{4}$

Given $\sinh{(x)}\sinh{(y)}=1$, I have to find the integral: $$\int_{0}^{+\infty}y\,dx=\dfrac{\pi^2}{4}.$$ I try to use the fact that this $$(e^x-e^{-x})(e^y-e^{-y})=4$$ut i have no idea of how to ...
-3
votes
0answers
27 views

Calculate F(10) up to 5 decimal places of accuracy [on hold]

I'm stuck with the following integral below, anyone have any ideas? I believe $f(t)\,dt$ is a generic function. $$F(x)=\int_0^{x^2+1} f(t)\,dt$$ the integral part is just the integral of $f(t)\,dt$ ...
4
votes
1answer
35 views

Double Integrals Calculus 3

Use a double integral to find the volume of the solid in the first octant bounded by the surfaces: z = xy, z = 0, y = x and x = 1. I did $$\int_0^1 \int_0^x (xy) \ dy\ dx$$ $$= \int_0^1 ...
1
vote
1answer
38 views

Why is $\int\limits_{\gamma} \frac{1}{z-1} \neq 2\pi i$, $\gamma = \{z : \lvert z \rvert = 1\}$?

$\int\limits_{\gamma} \frac{1}{z-1}$ $\gamma = \{z : \lvert z \rvert = 1\}$ I use Cauchy's integral formula, which says $$\int\limits_{\gamma} \frac{f(z)}{(z-a)^{n+1}} = \frac{2\pi i}{n!} ...
1
vote
1answer
53 views

Evaluate $\int_{0}^{1} \frac {\ln x}{1-x^2} \mathrm{d}x $

I found this question in a reference book: $$\int_{0}^{1} \frac {\ln x}{1-x^2} \mathrm{d}x $$ Can Anyone give me Idea how do I begin solving this?
0
votes
0answers
17 views

On the hypothesis of the change of variable theorem

I´m studying the change of variable theorem for a function $f:\mathbb R^n \to \mathbb R$ and my teacher gave us the theorem as follows: Theorem: Let $f:A\subset \mathbb R^n \to \mathbb R$ be ...
0
votes
0answers
18 views

Calculus 3 Spherical coordinates: I'm not sure how to set this up.

find the volume of the region enclosed by the sphere x^2+y^2+z^2=324 and the cylinder (x-9)^2+y^2=81 by using spherical coordinates. I'm just not seeing how to convert this into a form where spherical ...