All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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9
votes
4answers
216 views

How to evaluate a certain definite integral: $\int_{0}^{\infty}\frac{\log(x)}{e^{x}+1}dx$

How can I show that: $$\int_{0}^{\infty}\frac{\log(x)}{e^{x}+1}dx=-\frac{\log^{2}(2)}{2}$$ EDIT: This is equivalent to showing that $\eta'(1)=-\ln2\gamma-\dfrac{\ln^2(2)}{2}$.
2
votes
3answers
82 views

Calculate $\int\frac{dx}{x\sqrt{x^2-2}}$.

The exercise is: Calculate:$$\int\frac{dx}{x\sqrt{x^2-2}}$$ My first approach was: Let $z:=\sqrt{x^2-2}$ then $dx = dz \frac{\sqrt{x^2-2}}{x}$ and $x^2=z^2+2$ $$\int\frac{dx}{x\sqrt{x^2-2}} ...
4
votes
2answers
81 views

Integrating $z^{2n}\cos(1/z)/(1-z^n)$ over a circle of radius $2$ around the origin

I'm stuck on the following integral computation: $$\int_C \frac{z^{2n} \cos (1/z)}{1 - z^n} \, dz,$$ where $C$ is a circle of radius $2$ around the origin. I tried making the substitution $u = ...
8
votes
1answer
51 views

Changing the order of integration without sketching?

When changing the order of double integrals, I have always relied on sketching the region. I have recently come across this example on MSE by @FelixMartin which seems to avoid visual-based reasoning, ...
1
vote
1answer
43 views

Finding the region of integration

Let $A$ be the directed line from $(-1,-1)$ to $(1,1)$ and $B$ be the curve which starts at (1,1) and moves along $x^2+y^2=2$ up to $(-1,-1)$. Let $C$ be the union of $A$ and $B$ and $R$ be the region ...
4
votes
3answers
115 views

Solution of $\frac{d^2y}{dx^2} - \frac{H(x) y}{b} = H(-x)$

Does the equation $$\frac{d^2y}{dx^2} - \frac{H(x)}{b} y = c H(x)$$ have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant? Update: What about the second step ...
0
votes
1answer
54 views

Integration of differential forms

I have just started to learn differential forms. Now, there is a concept of pulling integral back. I somewhat understood the procedure to do it. But, I don't understand why we do it and when to use ...
3
votes
2answers
42 views

Multivariable integral limitation proof

Please prove the following formula. $$ \lim_{n\to\infty}\int_0^1\cdots\int_0^1 \frac{n}{\sum_{i=1}^n x_i}dx_1\cdots dx_n = 2 $$
1
vote
0answers
24 views

Showing that $f\in C'(\mathbb{R^2},\mathbb{R})$

Let $$f(x,y)=xy\int_{x^2-y^2}^{x^2+y^2}e^{\cos(xyt)}dt.$$ Prove that $f\in C'(\mathbb{R^2},\mathbb{R})$. I'm not exactly sure how to approach this problem. Here's what I've tried: First I ...
1
vote
2answers
70 views

An intergral with variable upper limit

Let $$\psi \left(x \right)=\int_{0}^{x}\frac{\ln(1-t)}{t}dt,x\in (0,1).$$ Show $$\forall x\in (0,1), \psi\left(x \right)=?$$ I return the old variable $t$ by the substitution $s=ln(1-t)$,and then ...
0
votes
2answers
38 views

Why the most animal reproduction is exponential growth? [closed]

I make a mathematical model explain why the most animal reproduction is exponential growth.but I don't know there are anyone did the same work.
1
vote
1answer
42 views

Finding the area bounded by two curves when in terms of $x = y^2$?

I can't seem to figure this problem out. Find the area bounded by the curves $x=2y-y^2$ and $x=4-y^2$, in the first quadrant. I am having difficulties with graphing the equations and coming up ...
-1
votes
1answer
61 views

Some confusing and tough (for me) integrations [closed]

Can anyone please help me with these integrations : $\int_0^3$ $| x+1 |$ $dx$ $\int$ $(|x-2|+|x-1|+|x|+|x+1|+|x+2|)$ $dx$ $\int$ $|x|dx$ $\int$ $(e^{|x|}$ + $\ln x)$ $dx$ $\int_0^\pi$ ...
0
votes
1answer
90 views

Calculate the integral of $\sqrt{36\sin^2(2t)+6\cos^2(t)}$

During an arc length calculation I reached the following integral and I am having hard time calculating it: $$\int\sqrt{36\sin^2(2t)+6\cos^2(t)}\,dt=\sqrt{6}\int\cos t \sqrt{24\sin^2(t)+1}\;dt$$ ...
1
vote
2answers
25 views

Finding the volume of a region rotated about the y-axis.

I'm having trouble trying to find the volume of the region formed by $y = x^2-7x+10$ and $y = x+3$ rotated about the y-axis. I was able to graph it, but I'm having difficulty when trying to come up ...
1
vote
1answer
44 views

Series converges but term by term integration not valid?

Give an example of a series $\sum g_n$ of Lebesgue integrable functions on $\mathbb{R}$ that converges but for which term by term integration is not valid. This is last minute exam revision so I do ...
7
votes
1answer
72 views

Closed form for integral of integer powers of Sinc function

(Edit: Thank you Vladimir for providing the references for the closed form value of the integrals. My revised question is then to how to derive this closed form.) For all $n\in\mathbb{N}^+$, ...
3
votes
0answers
78 views

Is this proof correct? Divergence of $\int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \, \mathrm{d}x $

Problem: Show that $$ \int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \,\mathrm{d}x $$ diverges. I know that there are many questions in which this problem is solved, but I want to know if my ...
1
vote
2answers
28 views

Evaluating a polar double integral on the semi disc

The integral: $$\iint_D (x^2-y^2)\,dx\,dy$$ where $D$ is defined as: $$\{(x,y)\in \mathbb R^2 \mid x^2+y^2\le 1, x\ge 0\}$$ Context I have solved double integrals on quarter discs but this semi ...
2
votes
2answers
39 views

Banach Space: Open Unit Ball Totally Bounded?

Just to be sure: In an infinite dimensional Banach space the open unit ball cannot be totally bounded, right? The context is that I need this in order to find a lack in here...
2
votes
1answer
67 views

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.
1
vote
0answers
42 views

Differentiation with respect to the index of the summation notion?

$$\sum_{t=1}^k \binom{N-1}{t-1} \int[1-F(s)]^{N-1}[F(s)]^{t-1}g(s)\,ds$$ $k\in\mathbb Z ^+$ If I want to find out the effects of changing $k$ (comparative statics), what can I do? Differentiation ...
1
vote
2answers
28 views

Why is Romberg integration usually based on trapezoidal rule?

The wikipedia article on Romberg Integration says that it's simply Richardson Extrapolation applied to either the Trapezoidal Rule or the Midpoint Rule. I'm reading out of a couple of textbooks on ...
-1
votes
1answer
23 views

What is the integral of $\int_{\mathbb{N}} s d\mu$ where $\mu$ is the counting measure on $\mathbb{P}(\mathbb{N})$?

What is the integral of $\int_{\mathbb{N}} s d\mu$ where $\mu$ is the counting measure on $\mathbb{P}(\mathbb{N})$? What does it mean for $s$ to be integrable? 1. This is last minute exam revision. ...
0
votes
1answer
21 views

Integrals with an imaginary linear term in the argument of the exponent

in this entry on Wikipedia stays $$ ...
6
votes
2answers
136 views

Find the length of the curve $x^{2k}+y^{2k} =1$

I want to find an expression for length and find the limit $k\rightarrow \infty$ The answer is obviously 8, if we look at the graphs.
0
votes
1answer
34 views

How to calculate the cumulative density of a bump function

How would I go about calculating the integral $$\int_{1/4}^x \exp\left(\frac{-1}{1-16(t-\frac{1}{2})^2}\right)dt,$$ where we assume $\frac{1}{4} \leq x \leq \frac{3}{4}$? Thanks!
1
vote
1answer
33 views

The integral of $1/x$ from $n-1$ to $n$ is greater than $1/n$?

I want to prove that $$\int_a^n \! 1/x \, \mathrm{d}x$$ where $a$ is $n-1$ is greater than $1/n$. How to prove this? I know the integral equals to $$\log(n) - \log(n-1)$$ But how to proceed from here? ...
1
vote
0answers
29 views

Do we need $\mu, \nu$ to be $\sigma$-finite to show $\int fg \ d(\mu \otimes \nu) = \int f \ d\mu \int g \ d\nu$?

The problem statement: Let $(X, \mathcal F, \mu), (Y, \mathcal G, \nu)$ be $\sigma$-finite and $f \in \mathcal L^1 (\mu), g \in \mathcal L^1 (\nu)$. Show that $fg \in \mathcal L^1 (\mu \otimes ...
1
vote
2answers
22 views

Limiting variable in interval: Lebesgue Dominated Convergence

So I am pretty comfortable using the LDCT for definite integrals and summations, but I am looking at a problem that has the interval as a function of the limiting variable, i.e.: $$\lim_{n\to\infty} ...
1
vote
1answer
24 views

How to modify Gauss-Hermite quadrature rule when the weight function is slightly generalized

hope this is the right forum. Consider a slightly modified version of the Gauss-Hermite quadrature rule, where the weight function is not $\exp(-\frac{x^2}{2})$ as in the standard Gauss-Hermite rule, ...
8
votes
4answers
127 views

How to find the derivative of a function defined by an integral? Namely, $f(y)=\int_0^{y^2} e^{-x^2y^2}dx$

Find at each point of its domain the derivative of the function $f: \mathbb{R} \rightarrow \mathbb{R}$ $$f(y)=\int_0^{y^2} e^{-x^2y^2}dx$$ $$$$ Is the domain of the function $\mathbb{R}$ because of ...
0
votes
2answers
77 views

A Matrix Integral Equation

We have an integral equation on matrix. ${\Im(t)}=\Im(0)+\int_{0}^{t} \Im(s)[K(s)]_{ \times }ds \tag 1$ $[\hspace{.2cm} ]_{\times}$ is skew symmetric matrix with diagonals zero and is non ...
2
votes
0answers
29 views

Solving $n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt$

I have to solve $$ n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt $$ where $\psi(t)=(2\pi)^{-\frac{1}{2}}e^{-\frac{1}{2}t^2}$ is the density ...
2
votes
3answers
79 views

any other method for evaluating $\int\limits \frac{ x^2+x+1 }{ \sqrt{x^2-x+1} } dx$?

I tried below and its getting tedious : $\begin{align}\\ \int\limits \frac{ x^2+x+1 }{ \sqrt{x^2-x+1} } dx &= \int\limits \frac{(2x-1)+ x^2-x+2 }{ \sqrt{x^2-x+1} } dx \\~\\ &= \int\limits ...
-1
votes
2answers
74 views

Integration by parts. integrate $\ln(x^2-x+2)$ [closed]

Integrate by parts $\ln(x^2-x+2)$
0
votes
3answers
37 views

How to find the arc length of this graph?

Could you please help me with this problem? Find the arc length of the graph of $y = \frac{x^{3}}{3} + \frac{1}{4x}$ between $x = 1$ and $x = 2$. Note: It may be helpful to use identities like $$x^{2} ...
1
vote
2answers
79 views

Find $ \int_0^2 \int_0^2\sqrt{5x^2+5y^2+8xy+1}\hspace{1mm}dy\hspace{1mm}dx$

I need the approximation to four decimals Not sure how to start or if a closed form solution exists All Ideas are appreciated
4
votes
2answers
102 views

Integral $\int^1_0\frac{\ln{x} \ \mathrm{Li}_2(x)}{1-x}dx$

I would like to know how to evaluate the integral $$\int^1_0\frac{\ln{x} \ \mathrm{Li}_2(x)}{1-x}dx$$ I tried expanding the integrand as a series but made little progress as I do not know how to ...
1
vote
1answer
45 views

How to understand uniform integrability?

From the definition to uniform integrability, I could not understand why "uniform" is used as qualifier. Can someone please enlighten me?
1
vote
0answers
31 views

Integral involving complimentary error function.

Essentially, I am trying to work out an integral of the form: $\int_{0}^{\pi }{x\cdot a\cdot \cos \left( x \right)e^{-\left[ a\cdot \sin \left( x \right) \right]^{2}}\mbox{erfc}\left[ -a\cdot \cos ...
0
votes
1answer
34 views

Trouble with integration using the definition of integral

I'm playing with integration for the first time and I can understand now why everyone tells me calculus II is the hardest calculus. I'm trying to solve this problem but I think I have the wrong ...
0
votes
0answers
31 views

What does the x represent in my integral

I am working on a project in which I have to write a matlab code in order to find the effective properties of a material with a super-spherical Inclusion. In order to do this, a paper I found, by ...
7
votes
2answers
91 views

Inequality of integrals $\int_0^1(f(x))^2 dx \geq 4$ if $\int_0^1xf(x) dx=\int_0^1f(x) dx = 1$

If $$\int_0^1xf(x) dx=\int_0^1f(x) dx = 1$$ prove that $$\int_0^1(f(x))^2 dx \geq 4$$ EDIT My attempt is as follows - I can use only the $\int xf(x)$dx part to get a bound $\int f^2(x) dx \geq ...
4
votes
0answers
87 views

Is it possible to find $\int \frac{1}{\sqrt[4]{1+x^4}} dx$ by parametrizing the curve $y^4-x^4=1$?

I found this integral in a handbook of integrals: $$\int \frac{1}{\sqrt[4]{1+x^4}} dx$$ I already have evaluated this integral by trigonometric substitutions and my answer agrees with that of the ...
0
votes
1answer
18 views

Change of variables theorem in the case $L^1_{\textrm{loc}}(U)$?

I'm trying to write a version of the change of variable theorem for the case of locally integrable functions on open subsets of $\mathbb R^n$. Statement: Let $U, V\subseteq \mathbb R^n$ be open sets ...
1
vote
2answers
47 views

Constants for anti-derivatives

Hey StackExchange I'm diving into integral calculus for the first time and I have a few questions about this problem. A steel ball bearing at rest is accelerated in a magnetic field in a line with ...
0
votes
1answer
42 views

How to find the area where $\frac{1}{z^2-4}$, $z \in \mathbb{C}$ is holomorphic?

Suppose that you are given a problem of finding the following complex integral: $$\int_\tau \frac{1}{z^2-4} dz$$ where $\tau = \{z \in \mathbb{C}: |z|=4 \}$. My question is (in the context of this ...
0
votes
1answer
52 views

Evaluating $\int{(a^2-x^2)^n}dx$ using repeated integration by part

The problem is as follows: Prove that $$\int{(a^2-x^2)^n}dx=\frac{x(a^2-x^2)^n}{2n+1}+\frac{2a^2n}{2n+1}\int{(a^2-x^2)^{n-1}}dx+C$$ using integration by part. I can easily obtain partially ...
0
votes
3answers
47 views

Calculate complex integral with pole at zero

Calculate for $\alpha >0$ and $n \in {\mathbb Z}$. $$ \oint_{\left\vert\,z\,\right\vert\ =\ \alpha} z^{n}\,{\rm d}z. $$