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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1
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2answers
49 views

Proving that two branch cuts can cancel out

Define the following functions $\mathbb{C}\to\mathbb{C}:$ $$u(z)=\frac{\log \left(z+\frac{1}{2}\right)}{z}\quad \left[-\pi\leqslant\arg \left(z+\tfrac12\right)<\pi\right];\quad v(z)=\frac{\log ...
0
votes
0answers
12 views

Trigonometric integral $\int_{[-\pi,\pi]^2}{\frac{1-e^{-in\cdot\theta_1}}{1-\cos(\theta_1)\cos(\theta_2)}\,d\theta_1\,d\theta_2}$

I am trying to compute the following integral (see here). Since it seems to be the wrong approach, I am trying to calculate another one which I hope it will give me what I am looking for. My point is ...
-2
votes
0answers
24 views

How to integrate $\int \left|f(x) +g(x) \right|^2dx$?

I am dealing with a quantum mechanics exercise at which I need to find the probabilty of $\left| \psi \right |^2$. $\psi$ is composed of 2 real value functions, say $f$ and $g$. So generally, how to ...
1
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3answers
23 views

Showing the integral $\int_{\mathbb{R}} \int_{\mathbb{R}} \min\{ 1, (\max \{ |x|,|y| \})^{-3} \} dx dy$ converges

I am trying to bound the following integral: $\int_{\mathbb{R}} \int_{\mathbb{R}} \min\{ 1, (\max \{ |x|,|y| \})^{-3} \} dx dy$. I am very sure this integral converges, but whatever I try seem to ...
5
votes
3answers
93 views

Integrate $\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$

I can't solve the integral $$\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$$ I tried it by using Beta and Gamma function and integration by parts. Please help me to solve it.
2
votes
1answer
33 views

If $f:[a,b]\rightarrow R$ is a uniformly continuous function then its absolutely continuous?

If $f:[a,b]\rightarrow R$ is a uniformly continuous function then is it true that $f$ is always absolutely continuous?
0
votes
2answers
63 views

Solve integral $\int \frac{x+1}{x^2-2x+5} dx$

I need to solve: $$\int \frac{x+1}{x^2-2x+5} dx$$ I cann see that $D>N$ so I tried to scompose the $D$ but I get: $$x_{1,2} = \frac{2 \pm \sqrt{4-20}}{2}$$ So $\Delta < 0$ and I tried to use ...
3
votes
1answer
16 views

What is an example of a uniformly continuous function but not absolutely continuous

Is there a function that is uniformly continuous function but not absolutely continuous. My answer is $f(x)=x^{2}, \forall x\in R$ Is this right? Are there any other?
1
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0answers
12 views

Fix a typo involving the Lobachevsky function in Thurston's notes

I believe that there is a typo in these great notes Thurston's Three-Dimensional Geometry and Topology, Volume 1 (Princeton University Press, 1997), Chapter 7 that is provide us by MSRI, in the ...
1
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0answers
25 views

Sum of all sine harmonics

I was discussing this with my calculus teacher, but she didn't come up with anything. I would like to take an infinite sum of functions (sine specifically) but don't know how to do that. I would ...
11
votes
1answer
248 views
+100

Why does the hard-looking integral $\int_{0}^{\infty}\frac{x\sin^2(x)}{\cosh(x)+\cos(x)}dx=1$?

I have to ask this question; most looking complicated definite integral yield not so nice closed form or irrational numbers or mixed of what ever ect. Why is this particular hard looking integral ...
0
votes
3answers
39 views

Domain of Integral $\int_{5}^{x} \frac {dt}{(1-t^2)}$

A function reads $$ F(x) = \int_{5}^{x} \frac {dt}{(1-t^2)} $$ Barrons says that the domain of F must be that $x >1$. But why can't $x$ be less than $1$ as well? As long as $x$ does not equal ...
2
votes
1answer
93 views
+50

Find the smallest value of the function $F:\alpha\in\mathbb R\rightarrow \int_0^2 f(x)f(a+x)dx$

Let $f -$ fixed continuous on the whole real axis function which is periodic with period $T = 2$, and it is known that the function $f$ decreases monotonically on the segment $[0, 1]$ increases ...
0
votes
1answer
25 views

Find $\int_{C}{\bf{F}}\cdot d{\bf{s}}$ through the line segment

Let $F=\left[\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}\right].$ Let $C$ by the curve consisting of the line segments from $$(-1,0)\to (0,-2)\to (2,0)\to (3,4)\to (0,5)\to (-1,0)$$ Find ...
4
votes
1answer
916 views

Leibniz's Rule for differentiation under the integral.

If we have $$F\left( \alpha \right) = \int\limits_a^b {f\left( {\alpha ,x} \right)dx} $$ Then $$\frac{{F\left( {\alpha + \Delta \alpha } \right) - F\left( \alpha \right)}}{{\Delta \alpha }} = ...
0
votes
0answers
10 views

$\int(\int\phi(a-z)dz)dz=\Phi(a-z)$

Lets assume $\phi(a-z)$ is integrable. Can I conclude that the following integral $$\int\left(\int\phi(a-z)dz\right)dz$$ Can be expressed by a function $$\Phi(a-z).$$ So in result: ...
3
votes
3answers
42 views

Why is $\frac{1}{x}$ not Lebesgue integrable on $[0,1]$?

My teacher said (without explaining) that $\frac{1}{x}$ is not Lebesgue integrable on $[0,1]$? Could someone please explain why is this true?
3
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4answers
74 views

Prove that $\int_{0}^{\frac{\pi}{2}}{\frac{\sin(2n+1)x}{\sin(x)}dx}=\frac{\pi}{2}$ for $n\ge0$

Prove that $$\int_{0}^{\frac{\pi}{2}}{\frac{\sin(2n+1)x}{\sin(x)}dx}=\frac{\pi}{2}$$ for $n\ge0$ I am not able to proceed with the integral. For the case $k+1$ please guide me through the problem. ...
91
votes
25answers
8k views

Is there any integral for the Golden Ratio?

This is a curiosity. I was wondering about math important/famous constants, like $e$, $\pi$, $\gamma$ and obviously $\phi$. The first three ones are really well known, and there are lots of integrals ...
1
vote
1answer
23 views

Triple integral $\int_{0}^{2\pi} \int_{0}^{2\cos(\theta)} \int_{0}^{\sqrt{2r\cos(\theta)}} r \ dzdrd\theta$ to find volume of a solid

On evaluating the volume between $$x^2+y^2 = 2x\\z^2=2x$$ I set up the triple integral $$\int_{0}^{2\pi} \int_{0}^{2\cos(\theta)} \int_{0}^{\sqrt{2r\cos(\theta)}} r \ dzdrd\theta$$ for which ...
4
votes
3answers
69 views

Limit solved by definite integral (Demidovich)

I was solving this limit from the Demidovich's book of exercises: $$\lim_{n\to\infty} \frac{\sqrt[n]{\vphantom{\Large a}\, n!\,}}{n}$$ and I managed to get it to this state but then I got stuck: ...
3
votes
5answers
142 views

Evaluating the definite integral $\int_0^3 \sqrt{9- x^2} \, dx$

I have been having a problem with the following definite integral: $$\int_0^3 \sqrt{9- x^2} \, dx $$ I am only familiar with u-substitution and am positive that it can be done with only that. Any ...
-1
votes
0answers
12 views

Estimation for expression $\frac{{\int {{{\left( {h'\left( x \right)} \right)}^2}dx} }}{{\int {{{\left( {h\left( x \right)} \right)}^2}dx} }}$

Problem. Find an estimation for $C$ in the following integral inequality: $\frac{{\int\limits_a^b {{{\left( {h'\left( x \right)} \right)}^2}dx} }}{{\int\limits_a^b {{{\left( {h\left( x \right)} ...
1
vote
0answers
75 views

How to solve this definite integral $\int_0^\pi \frac{\cos^9(x)}{\sin^3(x)+\cos^3(x)}dx$

I'm having trouble evaluating the following integral: $$ \int^\pi_0 \frac{\cos^9(x)}{\sin^3(x)+\cos^3(x)}dx $$ I tried to convert it into an algebraic function by multiplying the numerator and ...
2
votes
1answer
53 views

Evaluate definite integral using the definition: $\int_{-3}^{1}(x^2-2x+4) dx$?

Can someone walk me through finding the value of a definite integral by using the definition itself? In this case: `The definition of the definite integral: let $f$ be a function that is defined on ...
0
votes
1answer
55 views

how to evaluate this definite integral $\int_0^\infty\frac{\sin^2(x)}{x^2}dx$? [duplicate]

For $\int_{0}^{\infty}\frac{\sin^2(x)}{x^2}dx$. I considered using residue theorem. But since the function inside is holomorphic except for a removable singularity at the origin. So whatever contour I ...
2
votes
2answers
397 views

How to evaluate the definite integral by the limit definition $\int_{-1}^1 x^3 dx$?

Solve the definite integral by the limit definition: $$\int_{-1}^1 x^3 dx$$ The formula: $$\int_a^bf(x)dx= \lim_{n\rightarrow \infty} \sum_{i=1}^n f(c_i)\Delta x_i$$ Get the variables: $$\Delta ...
4
votes
2answers
190 views

Evaluating a definite integral by definition: $A(x) = \int_{-1}^x (t^2 + 1)\space dt$

I have an area function $A(x)$ defined as $$A(x) = \int_{-1}^{x} (t^2 + 1)\space dt$$ ... and I would like to use the definition of definite integral to evaluate it. I started this way $$A(x) = ...
1
vote
3answers
88 views

How should I try to evaluate this integral $\int_0^\pi \sqrt{1+4\sin^2\frac x2 - 4\sin\frac x2}\;dx$?

Suppose that we are given the following integral: $$\int_0^\pi \sqrt{1+4\sin^2\frac x2 - 4\sin\frac x2}\;dx.$$ (Original screenshot) And the answer is one of these :- $4\sqrt3-4-\frac\pi3$ ...
2
votes
1answer
80 views

How to evaluate this definite integral $ \int_{-\pi/3}^{\pi/3} \frac{(\pi +4x^3)\,dx}{2-\cos(|x|+ \frac{\pi}{3})} $

$$ \int_{-\pi/3}^{\pi/3} \frac{(\pi +4x^3)\,dx}{2-\cos(|x|+ \frac{\pi}{3})} $$ I have separated the integral into two parts, then expanded using $\cos(a+b)$ formula, after that I am lost. Can ...
0
votes
2answers
69 views

Evaluating definite integral $\int_ 3^6 \frac1{\sqrt{27+6x-x^2}} dx$

Evaluate the following definite integral. $ \int _{ 3 }^{ 6 }{ \frac { 1 }{ \sqrt { 27+6x-{ x }^{ 2 } } } } \quad dx $ I can't figure out how to complete the square.
0
votes
1answer
50 views

Sophomores dream

On wiki there is a proof of Sophomore's dream. I am trying to understand what they did when changing the variable and how they got $e^{-u}$.
2
votes
1answer
59 views

Show that $\int h_n'(x) \varphi(x)\, dx \to \langle \delta, \varphi\rangle$ - Generalized functions theory

In the book (EDIT) of Robert Strichartz, there's an exercise (#$1$, page $9$) that I am not quite sure how to solve. Is there anyone could give me the principal steps how to solve it? Problem : Let ...
3
votes
2answers
89 views

Possible to do better than an upper bound for$\int^{\infty}_0 e^{-x}\log(x)\ dx$?

I used the series expansion of $e^{-x}$ and the fact that $\log(x)$ was less than $x$ in the $(0, \infty)$ to get an upper bound and so use simple comparison to show this was indeed integrable over ...
35
votes
8answers
41k views

Why is the area under a curve the integral?

I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself. However when it comes to the area ...
2
votes
2answers
75 views

Computation of $\int _{-\pi} ^\pi \frac {e^{in\theta} - e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$

I need to compute $$\int \limits _{-\pi} ^\pi \frac {e^{in\theta} - e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$$ Does anyone see any good strategy? Thanks.
8
votes
4answers
145 views

How does one integrate $x^2 \frac{e^x}{(e^x+1)^2}$?

How can I show this? $$ \int_{-\infty}^{\infty} x^2 \frac{e^x}{(e^x+1)^2} dx = \pi^2/3$$ I tried applying residuals, but the pole is of infinite(?) order.
0
votes
1answer
29 views

Finding a line integral of $5ydx+7xdy$.

Evaluate the line integral $$\int_C5ydx+7xdy $$ where $C$ is the straight line path from $(2,3)$ to $(7, 5)$. I don't know the way to do that I tried many ways but I still could not get the right ...
-1
votes
2answers
62 views

show that $\int_{0}^{1}\frac{1}{x^2}\ln\left[\frac{(1+x^2)^2}{1-x^2}\right]dx=\pi$

Most integrals involved $\ln(x)$ seem to produced results of $\pi^2$, $\sqrt\pi$, $\pi\ln(2)$ etc, but rarely $\pi$ on its own. Here is one (1) ...
0
votes
1answer
42 views

Prove $\int_{0}^{1}\frac{1}{x^2}\ln\left[\frac{(1+x^2)^2}{1-x^2}\right]dx=\pi$

Most integrals involved $\ln(x)$ seem to produced results of $\pi^2$, $\sqrt\pi$, $\pi\ln(2)$ etc, but rarely $\pi$ on its own. Here is one (1) ...
0
votes
1answer
21 views

Enlargement of area and perimeter in a rotation body

Let $f: [0,1] \to \mathbb{R}$ a continuous, differentiable function with $f \ge 0$. Rotate the graph of $f$ around the x-axis. Define this rotation body in $\mathbb{R}^3$ with $A$ and the area in ...
-1
votes
1answer
29 views

How to solve simultaneous inequalities (reasked)? [duplicate]

I am doing multivariable calculus, and specifically double integrals. I am facing difficulties finding the domain of the integal, however i am given the following equations: $$1≤2x+y≤2$$ $$0≤x−2y≤1$$ ...
0
votes
1answer
18 views

Doubts in Volume, Hypervolume in $R^4$

Recently I was reading about triple integrals and I came across the statement - "We saw that a double integral could be thought of as the volume under a two-dimensional surface. It turns out ...
1
vote
1answer
40 views

Prove $\int_{0}^{1}\int_{0}^{1}[-\ln(xy)]^s\left(\frac{1}{\ln(xy)}+\frac{1}{1-xy}\right)dxdy=\Gamma(s+2)\left[\zeta(s+2)-\frac{1}{s+1}\right]$

I got the idea from here $(1)$ yield the same result as Hadjicostas. Is this $(1)$ same as Hadjicostas but just write in a different way? ...
0
votes
0answers
29 views

Need help integrating $\frac{1}{160}\log \left(5x-25\right)\left(\left(y-146\right)^2+\left(x-7\right)^2\right)=10$

The project is quite simple that I am making much more difficult because it is fun to make things difficult. Create a single function for a water tower design with a narrow part and reservoir at least ...
5
votes
2answers
203 views

Calculate the Gauss integral without squaring it first

We know that the integral $$I = \int_{-\infty}^{\infty} \mathrm{d}x e^{-x^2}$$ can be calculated by first squaring it and then treat it as a $2-$dimensional integral in the plane and integrate it in ...
-1
votes
0answers
13 views

Integral becoming anti derivative [on hold]

How does integral become anti derivative? I know the proof of it.I want to make the concept clear through graph or some nice simple sentences
1
vote
1answer
105 views

Why does the hard-looking double integral $\int_{0}^{1}\int_{0}^{1}\frac{x(1-x)y(1+y)}{(1-xy)\ln(xy)}dxdy=-\frac{1}{2}$?

(1) originate from here problem 11322 (1) $$\int_{0}^{1}\int_{0}^{1}\frac{x(1-x)y(1+y)}{(1-xy)\ln(xy)}dxdy=-\frac{1}{2}$$ On my recent post see here Marco Cantarini and hints from other proved the ...
0
votes
1answer
20 views

Help Computing integral of continuous function

Consider the Dirichlet function $F:R \rightarrow R$ given by $f(x):=\begin{cases} x &\text{if } x < 1 , \\{}\\ x+1 &\text{if } 1<= x <= 2, \\{}\\ -x+5 &\text{if } 2<x ...
0
votes
0answers
22 views

problem solving these differential equations

$$(\sin^2(x) D^2 +\sin(2x) D+\cos^2 x+1)y=\sin^3 x$$ and there is another $$(xD^2-x(x+2)D+(x+2))y=x^3-2x+1$$