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 ...

learn more… | top users | synonyms (3)

0
votes
1answer
10 views

Integrability of a function

Show that the function is integrable on [0,2] $$f(x)=\left\{\begin{array}{cc} 1-x & x<1 \\ x^2-2x+1 & x\:\ge 1 \end{array}\right.$$ What conditions need to be checked in order for it to be ...
0
votes
0answers
26 views

Is it possible to integrate this Riemann zeta function ratio so that I can produce this graph?

I am partly repeating myself here. But the form of this expression is nicer than the one I suggested here. I would like to integrate this: $$1-\frac{\zeta \left(\frac{1}{2}+i t\right)}{\zeta ...
1
vote
1answer
30 views

Integration of a generic radial function in polar coordinates

I need to perform the following integral $\int{P(k) e^{i \vec{k}\cdot \vec{\Delta r}} \frac{d^2k}{(2 \pi) ^2}}$ using polar coordinates. I think the result should depend on some Bessel function, but ...
1
vote
0answers
23 views

Convergence of a sequence of integration

I am considering one problem and I am stuck in this step. The problem is that What conditions on function $f(u,\epsilon)$ are required to satisfy $$ \int_0^\epsilon f(u,\epsilon)\,du \rightarrow 0 ...
0
votes
3answers
34 views

Evaluating the closed integral of an elliptical path

I've been working on a problem that states: Evaluate $\int F*dr $ where $F(x,y,z) = x\,i+xy\,j+x^2yz\,k $ and C is the elliptical path given by $$ x^2+4y^2-8y+3=0 $$ in the xy-plane, traversed ...
0
votes
3answers
74 views

I have great doubts solve this exercise by integral by parts $\int_{0}^1 \int_0^1 x\cdot e^{xy}\, dy\, dx$ [on hold]

I have great doubts solve this exercise by integral by parts $\int_{0}^1 \int_0^1 x\cdot e^{xy}\, dy\, dx$
0
votes
1answer
15 views

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis

I am having a little trouble figuring out how to integrate this problem. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis. ...
0
votes
1answer
20 views

Finding the surface area of the solid formed by a revolution of the function $f(y)=x$ when rotated about the line $y=0$.

I know of the following formulas for calculating surface areas: $\displaystyle A_S = 2\pi\int_{a}^{b}f(x)\sqrt{1+f'(x)^2}{\ dx}$ for the surface area ($A_S$) of the solid formed by revolving $f(x) = ...
0
votes
1answer
29 views

Lebesgue-Stieltjes: Computation

Problem Given the real line $\mathbb{R}$. Consider a Borel family: $$\mu(\mathbb{R})<\infty:\quad\mu(\lambda):=\mu(-\infty,\lambda]$$ How can I compute: ...
2
votes
2answers
91 views

How to calculate $\int \frac{\sin x}{\tan x+\cos x} \, dx$

How to calculate $$\int \frac{\sin x}{\tan x+\cos x} \, dx\text{ ?}$$ I got to $$\int \frac{-u}{u^2-u-1} \, du$$ while $u=\sin x$ but can I continue from here?
2
votes
1answer
44 views

Assumptions on functions so that integral is zero

Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be two arbitrary functions. Assume $g\in L^2(\mathbb{R})$. I'm looking to find out the minimal set of assumptions on $f$ and $g$ such ...
1
vote
1answer
31 views

find the minimum value of this integral when $1>t>0$, $f(t)=\int\limits_0^1 x |e^{-x^2} - t|\ \mathrm{d}x = ?$

Is there someone who can show me How do i find the minimum value of this integral when $1>t>0$, \begin{align*}f(t)=\int\limits_0^1 x |e^{-x^2} - t|\ \mathrm{d}x &= \end{align*} Note : ...
0
votes
1answer
40 views

convolution and integral limits

Let $\xi$ be an increasing function , and $f$ be a continuous function on the interval $[0,1]$. Take $\phi$ a smooth function such that $\int_0^1 \phi(s)\, ds= 1 $ and consider an approximation of ...
0
votes
1answer
19 views

Asymptotic behaviour of Hilbert transform

Let $f$ be a bounded function on $\mathbb{R}$ with compact support include in $[-K,K]$. Show that $$ H(f)(x)=\frac{a}{\pi x}+O(\frac{1}{x^2})$$ where $a=\int f(t)dt$ and $H$ denote the Hilbert ...
0
votes
0answers
25 views

Change of variable in double and triple integrals?

I learn double and triples integral as same as change of variable and then surface integral in my class so there is some conflict between how to do double integrals Here is how the text book say ...
2
votes
3answers
63 views

Does $\int_a^\infty f$ exist iff $\int_a^\infty |f|$ exists?

My question is, does $\int_a^\infty f(x)dx$ exist if and only if $\int_a^\infty |f(x)|dx$ converges? Since $$\left|\int_a^\infty f(x)dx\right|\leq \int_a^\infty |f(x)|dx,$$ it's obvious that if ...
3
votes
1answer
55 views

Does $\int_0^\infty \frac{1}{1+(x\sin x)^2}\ dx$ converge?

Does the integral $$\int_0^\infty \frac{1}{1+(x\sin x)^2} \ \, \mathrm{d}x$$ converge? I know that I need to look at: $$\sum_{n=0}^\infty \int_{n\pi}^{(n+1)\pi} \frac{1}{1+(x\sin x)^2}\ \, ...
3
votes
1answer
34 views

Proving that the Gamma function $\Gamma(y)$ converges for $y>0$.

How can I justify that $$\Gamma(y)=\int_0^\infty t^{y-1}e^{-t} \, \mathrm{d}t$$ exists for all $y>0$? I'm struggling to compare it to a known convergent integral.
0
votes
1answer
27 views

Every step function is a linear combination of elementary step functions.

If $J$ is any subinterval of $[a, b]$ and if $\phi_J (x) := 1$ for $x \in J$ and $\phi_J (x) := 0$ elsewhere on $[a, b]$, we say that $\phi_J$ is an elementary step function on $[a, b]$. Then to ...
1
vote
2answers
36 views

Proving $\int_0^1 \frac{f(t)}{t^{\alpha + 1}} \ dt$ diverges

Consider $f(t)$, continuous on $[0,1]$, and $\alpha > 1$, and: $$\int_0^1 \frac{f(t)}{t^{\alpha + 1}} \ dt$$ How can we tell this integral diverges? Basically since $f$ is continuous it reaches ...
0
votes
0answers
32 views

Is my proof of closedness of multiplication operator corect?

I am considering an operator $A: L^2(\mathbb R , d \mu) \supset D(A)\to L^2 (\mathbb R, d\mu)$ defined by $(Af)(x)=a(x)f(x)$ for known measurable function $a$. Domain is of course all those functions ...
0
votes
6answers
72 views

How to evaluate this integral $\int\limits_1^4\!\left( \frac{1}{\sqrt{x}}+\frac{1}{x}\right) \mathrm{d}x $?

$$\int_1^4\!\left( \frac{1}{\sqrt{x}}+\frac{1}{x}\right) \mathrm{d}x $$ The answer is $2+\ln(4)$, however I don't understand why. What I did was the following: $$\ln(x^{0.5})+\ln(x) = ...
2
votes
1answer
37 views

Help with Definite integral question

Anyone please help with this question: (a) Show that: \begin{align} \int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx \end{align} (b) Hence show that: \begin{align} \int_{0}^{\frac{\pi}{4}} ...
4
votes
4answers
406 views

Why can we treat infinitesimals as real numbers in integration by substitution?

During integration by substitution we normally treat infinitesimals as real numbers, though I have been made aware that they are not real numbers but merely symbolic, and yet we still can, apparently, ...
1
vote
1answer
45 views

Finding the general integrals of functions like $\frac1{x^n+1}$, $\cos^nx$. [on hold]

This question is just a soft question, about can we compute a general formula for everything? Or it has some restrictions? Like $\int x^ndx=\frac{x^{n+1}}{n+1}+C$. I am not able to deduce a formula ...
1
vote
1answer
45 views

Help understanding proof on Jensen's Inequality

I need help understanding the proof for Jensen's inequality in "Real and Complex Analysis" by Rudin. 3.3 Theorem (Jensen's Inequality) Let $\mu$ be a positive measure on a $\sigma$-algebra ...
2
votes
3answers
128 views

How do i evaluate this integral $ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $?

Is there some one show me how do i evaluate this integral :$$ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $$ Note :By mathematica,the result is : $\frac{Gamma\left(\frac1 ...
2
votes
3answers
46 views

Continuity of function consisting of an infinite series.

Let $f(x) , 0\leq x\leq 1$ be defined by, $$f(x)=\sum_{n=1}^{\infty}\frac{1}{(x+n)^2}$$. Show that $f$ is continuous on $[0,1]$ and that, $$\int_0^1f(x)dx=1$$. I have never dealt ...
0
votes
0answers
24 views

Searching for a condition on the derivative $f_u$

Please wht can be the condition on $f_u$ such that we obtain the following equality: $$\int_0^1 \int_0^1 G(t,s)f_u(s,0) v(s) w(t) \ ds\ dt=\int_0^1 \int_0^1 G(t,s)f_u(s,0) w(s) v(t) \ ds\ dt$$ ...
0
votes
1answer
53 views

For what values of $a$ does $\int_{0}^{1}(-\ln x)^adx$ converge?

For what values of $a$ does $\int_{0}^{1}(-\ln x)^adx$ converge? I have seen a duplicate of this question but the answer there, though very good and creative, isn't clear about negative values. When ...
-2
votes
2answers
79 views

How to evaluate $\int \frac{\mathrm dx}{1+\sin x−\cos x} $?

Is there someone show me how I evaluate this integral:$$\int\frac{\mathrm{d}x}{1+\sin x−\cos x} $$ I used $t=\tan\frac{x}{2}$ but i didn't succeed . Thank you for any help .
1
vote
2answers
46 views

How to show the integral $\int_e^\infty \left(\frac{e}{t}\right)^t dt$ converges?

Let $$I=\int_e^\infty \left(\frac{e}{t}\right)^t dt$$ How to show it converges? I tried to find some inequality to compare with.
1
vote
4answers
108 views

Integral of $\frac{x^2+1}{(1-x^2)\sqrt{1+x^4}}$ [duplicate]

So we have to evaluate $\int\frac{x^2+1}{(1-x^2)\sqrt{1+x^4}}dx$. My work- We can write the integrand as $\frac{(x+1)^2-2x}{(1-x)(1+x)\sqrt{1+x^4}}dx$. So we wish to deduce ...
-3
votes
0answers
58 views

How would you show that $\lim_{n \to \infty} (1+ \frac{1}{n})^n$ is equal to $e$? [duplicate]

How would you show that $\lim_{n \to \infty} (1+ \frac{1}{n})^n$ is equal to $e$?
1
vote
2answers
56 views

Show there exist a constant $c\in \Bbb{C}$ such that $\int_{0}^{1}|{f-c}|^2<{1\over 36}$

Let $f:\Bbb{R}\to \Bbb{C}$ be a $1$-periodic function, $f\in C^1$ and $\int_{0}^{1}|f'|^2\le 1$. a. Show $\sum_{k\ne 0}|{\hat{f}(n)}|^2\le {1\over 4\pi^2}$ (I did it already, and that question is ...
0
votes
6answers
50 views

Integral of $ \frac{dx}{\sqrt{x^2 + 1}} $ ( and other table integrals ) [duplicate]

I am wondering how to prove this integral: $$ \int \frac{dx}{\sqrt{x^2 + 1}} $$ Of course, i know the solution to this integral, since it's one of the table integrals i.e. $$ \int ...
1
vote
1answer
33 views

Change of order of integration of a triple integral

Consider $$ I = \int_0^{\omega}\int_0^{\alpha}\int_0^{\alpha}F(\beta){\tilde{F}(\gamma)}e^{i\beta t}e^{-i\gamma t}R(\alpha)d\beta d\gamma d\alpha$$ In this triple integral,I want to bring about, a ...
-1
votes
0answers
18 views

How to solve this double integral problem? [on hold]

$$D: y \leq 1, x^2 \leq y$$ $$\iint_D (y+yxf(x^2+y^2))\,dx\,dy$$
0
votes
1answer
41 views

Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and $h=f-g$

Please please please please please I want some help ,Is there and body here who can help me in this question : Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and ...
1
vote
0answers
42 views

Find the hydrostatic force using integration

A vertical dam has a semicircular gate. Find the hydrostatic force against the gate. The dam is 12 meters high, the water level is at 10 meters, and the semicircular gate had a diameter of 4 meters. ...
0
votes
1answer
75 views

evaluating $ \int\limits _{0}^{1}\frac{1}{\sqrt{x+\varepsilon}}dx $

I came across this : I'm trying to evaluate it up to $ o(\epsilon) $ $$ F\left(\varepsilon\right)=\int\limits _{0}^{1}\frac{1}{\sqrt{x+\varepsilon}} \, \mathrm{d}x $$ I've trying considering to look ...
1
vote
1answer
79 views

Show that $\lim_{x\to\infty} f(x) = 0$.

Let $f\in C^1$. Let's assume that $\int_0^\infty f(x)\ dx$ converges and $f'(x)$ is bounded. Prove that $\lim_{x\to\infty} f(x) = 0$. Let's assume by contradiction that $\lim_{x\to\infty} f(x) ...
2
votes
1answer
197 views

Integration by substitution - where is the mistake?

I want to integrate $$\int_{-1}^{1} (1-x^2)^{3/2} \, \mathrm{d}x$$ by substituting $x=\cos z$ and $dx = -\sin z \, dz$. $x=-1 \implies z=-\pi $ and $x=1 \implies z=0$. I receive: ...
3
votes
4answers
147 views

Is integration of $x\operatorname{cosec}(x)$ defined?

Is integration of $x\operatorname{cosec}(x)$ possible? If yes, then what is its closed form; if not, then why is it non-integrable ?
2
votes
2answers
49 views

An improper integral and its convegence

I have an integral $$I(\gamma)=\int\int d^3 \mathbf{r} \, d^3 \mathbf{r}' \frac{1}{|\mathbf{r}-\mathbf{r}'|+\gamma}$$ were $\gamma$ is a positive number, $\mathbf{r},\mathbf{r}' \in \mathbb{R}^3$, ...
0
votes
1answer
31 views

Is the integral with respect to increasing continuous functions the limit of integrals with respect to $C^1$ functions?

if $\xi$ is continuous increasing can we find $\xi^n\in C^1$ such that $$\int_0^t f(u)\, d\xi = \lim_n\int_0^t f(u)\, d\xi^n$$ for every continuous $f$?
-2
votes
3answers
72 views

Integral of rational function with a squared term in the denominator

I know the integration when in the reciprocal there's only degree $1$, but what about degree $2$? Take an example, $$\int\frac{x \, \mathrm{d}x}{a+bx^2}$$
1
vote
1answer
22 views

Bounds for double exponential integrals

I understand that the double-exponential integral $$ F(a,b,C) := \int_{C}^\infty \exp(-a \exp(b x)) \, dx \quad \text{(with $a,b>0$ and $C \geq 0$)} $$ can in general not be solved in closed-form. ...
0
votes
2answers
48 views

Evaluating the integral in polar

I am trying to show that the double integral of $\sqrt{\rho^2-y^2}$ for $x$ between $0$ and $\rho$ while $y$ is between $0$ and $\sqrt{\rho^2-x^2}$ is $(2/3)(\rho)$. In cartesian I have tried its ...
0
votes
1answer
38 views

Area of region - double integral

Here is my task: Calculate area of region $(x^{2}+y^{2})^{2}\leq a^{2}(x^{2}-y^{2})$. Here is what I have done. After transforming this line to polar form $(x=\rho\cos\phi,y=\rho\sin\phi)$, we have: ...