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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4
votes
2answers
114 views

Evaluate $I = ∫∫ 1/((x^2 + y^2)^{n/2}) dxdy$

Evaluate the double integral $$ I = \int\int_D \frac{1}{(x^2 + y^2)^{n/2}} dxdy .$$ where $n$ is an integer and $D$ is the region of the plane bounded by two circles centered on the origin and ...
0
votes
1answer
152 views

How to calculate this integral in 3 dimensions involving the Dirac delta function?

How would I go about calculating the integral $ \int d^3 \mathbf r {1\over 1+ \mathbf r \cdot \mathbf r} \delta(\mathbf r - \mathbf r_0) $ where $\mathbf r_0 = (2,-1,3)$ My attempt so far: I have ...
0
votes
1answer
37 views

Integral and function.

I am given function $f(t) = \frac{1}{4\cdot\left(t-1\right)^{\frac{1}{3}}}$ I have to write this function (for $ x>0 $): $\phi (x)=\int_{0}^{x}f(t)= ?$ I calculate integral: ...
2
votes
1answer
119 views

Evaluating a trigonometric integral by means of contour $\int_0^{\pi} \frac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta$

I am studying for a qualifying exam, and this contour integral is getting pretty messy: $\displaystyle I = \int_0^{\pi} \dfrac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta $ I first notice that the ...
1
vote
1answer
254 views

Clayton copula and Kendall's tau

I'm currently preparing for an exam in Risk Management (mathematics) by doing exercises from old exams. One of these exercises proved to be too difficult because of the following: Given Kendall's tau ...
0
votes
0answers
40 views

Area of intersection between square and annulus

The annulus' larger radius is $1$, smaller radius is $r>0$, and center is $(0,0)$. The square's sides are parallel with the axes, the lower left corner's coordinates are $(a,b)$, and the upper ...
1
vote
2answers
59 views

Integral with several parameters

Let $r>0$. Find $(p,q) \in \mathbb{R}^{2}$ such that the integral: $$\int_{1}^{\infty}{\frac{(x^{r}-1)^{p}}{x^{q}}} ~dx$$ converges and for those values calculate it. I've already ...
0
votes
1answer
38 views

what are measurable spaces on the real line?

I've came across this article about the dominated convergence theorem , but since i didn't take a course on measure theory , i have some problems understanding the language of the previously or other ...
0
votes
2answers
192 views

Difference between summation and integration

It is well known that if a series $\sum\limits_{k= 0}^\infty a_k$ converges, then $a_k \to 0$. However, this is not true for integrals. What makes them different? Is it simply that they are ...
6
votes
3answers
421 views

How to better spot the right integration by parts

I was having trouble integrating $$ \int_0^{\pi/2}\sin^{n}\left(x\right)\cos^{2}\left(x\right)\,{\rm d}x $$ and someone pointed out to me that it was a somewhat simple integration by parts. Does ...
1
vote
3answers
103 views

$W_n=\int_0^{\pi/2}\sin^n(x)\,dx$ Find a relation between $W_{n+2}$ and $W_n$

Set $$W_n=\int_0^{\pi/2}\sin^n(x)\,dx.$$ Compute $W_0$ and $W_1$. Find a relation between $W_n$ and $W_{n+2}$ and deduce a formula for $W_n$. What I have so far is: $$W_{2k}=\frac{1}{2^k}\left( ...
4
votes
4answers
194 views

Using substitution to evaluate indefinite integral $ \int{x\sqrt{4x+1}}dx$

Evaluate this indefinite integral. $$I= \int{x\sqrt{4x+1}}dx$$ Let $u=4x+1$ $$\frac{du}{dx}=4\rightarrow{dx=\frac{du}{4}}$$ $$I=\int{x}\sqrt{u}\frac{1}{4}du=\frac{1}{4}\int{x}\sqrt{u}du$$ Then ...
2
votes
1answer
59 views

Invariance of integral

Given the Lebesgue integral with the Lebesgue measure and the Borel-Sigma Algebra, I am supposed to figure out under which transformations $\int_{\mathbb{R}^2} f(x) dx$ the integral is ...
0
votes
1answer
89 views

Is it possible to separate the variables in this equation without a substitution?

I have this: $$\frac{dy}{dx}=-\frac{4x}{y}-\frac{y}{x}$$ I want to separate it to solve for y. I could do it by subbing $v=\frac{y}{x}$, separating v and x, solve for v, then unsub v. But is ...
2
votes
0answers
53 views

Strategies for swapping the order of integration with dependent bounds

What are the general strategies for swapping the order of integration given dependent bounds? Specifically, in $\mathbb{R}^2$, Fubini's theorem allows us the following $$ \int_{a}^b\int_{c}^d ...
4
votes
2answers
62 views

Given integral $\iint_D (e^{x^2 + y^2}) \,dx \,dy$ in the domain $D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$ Move to polar coordinates.

Given integral $\iint_D (e^{x^2 + y^2}) \,dx \,dy$ in the domain $D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$ Move to polar coordinates. First of all I tried to find the domain of $x$ and ...
3
votes
1answer
156 views

How can one prove the impossibility of writing $ \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions?

Can we express $ \displaystyle \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions? (Note: Infinite series are not allowed.) If not, then is there a proof that $ \displaystyle \int ...
0
votes
0answers
21 views

Exchange Rate Model

I'm designing a model for a volume-dependent exchange rate system. Here is the model: There is an entity called the exchanger that holds reserves of every kind of currency. Lets say there are two ...
3
votes
2answers
105 views

Problem with integral.

How can I evaluate this integral? $$ \int{x^{3}\,{\rm d}x \over \left(x - 1\right)^{2}\sqrt{x^{2} + 2x + 4}}$$ I would be grateful for any tips.
2
votes
1answer
132 views

Integration of a Gaussian multiplied by a Bessel function

I currently have a hard time figuring out the following integral: Integrate[r*Exp[-r^2/h^2]*BesselJ[0,i*k*r/z],{r,0,a}] I've written it down in the Mathematica typeset and hope you can help me! ...
2
votes
1answer
76 views

Evaluating $\int\frac{ {\operatorname d}x}{1-x^3}$

How to find $$\int\frac{{\operatorname d}x}{1-x^3}?$$ Is it possible by sine or cosine functions? It is not easy to calculate it by reparametrization.
0
votes
0answers
41 views

Integration by substitution and transformation

Let $B_r \subset \mathbb{R}^{n-1}$ be a ball of radius $r$ centred at $0$. Let $h \in C^{0,1}(\overline{B_r})$. Consider $$I=\int_{B_r}u\big(\phi(y_1, ..., y_{n-1}, h(y_1, ..., ...
-3
votes
1answer
46 views

Prove convergence of this integration [duplicate]

$$\int_0^\infty\frac1{(x^2+1)(\ln^2(x)+\pi^2)}$$ Prove the convergence of this integration.
1
vote
1answer
85 views

munkres analysis integration question

Let $[0,1]^2 = [0,1] \times [0,1]$. Let $f: [0,1]^2 \to \mathbb{R}$ be defined by setting $f(x,y)=0$ if $y \neq x$, and $f(x,y) = 1$ if $y=x$. Show that $f$ is integrable over $[0,1]^2$.
3
votes
1answer
37 views

Integrability of time differences via bootstraping?

the question is somehow inspired by the Alt-Luckhaus paper (Lemma 1.5) Let $B:\mathbb{R}\to\mathbb{R}$ be continuos and nonnegative, $\Omega\subset \mathbb{R}^n$ a bounded domain, $h,T>0$. Let ...
0
votes
2answers
66 views

Compute integral $\int_{0}^{1} t^{i\eta}(1-tz)^{-2} \, \mathrm{d}t$ analytically

I need to compute hypergeometric function: $$_2F_1(1+i\eta, 2; 2+i\eta, z)$$ After applying the integral representation, the task is now to compute integral: $$\int_{0}^{1} t^{i\eta}(1-tz)^{-2} ...
3
votes
2answers
157 views

How find this integral $\int_{0}^{\pi}\dfrac{2t+2\cos{x}}{t^2+2t\cos{x}+1}dx$

Find this follow integral $$F(t)=\int_{0}^{\pi}\dfrac{2t+2\cos{x}}{t^2+2t\cos{x}+1}dx$$ where $t\in R$ my try: ...
1
vote
2answers
261 views

Integrating exponential of exponential function: stuck at integration by parts

I want to integrate $$\int_{0}^{t}\exp\left\{{k_{1}\left ( 1-e^{-t/{k_{2}}} \right )}\right\}dt$$ First I substituted $u = 1-e^{-t/{k_{2}}}$ Thus I get ...
5
votes
3answers
128 views

Evaluating $\int \frac{\operatorname dx}{x\log x}$

How to integrate $\frac{1}{x\log x}$? Could you give me some ideas on how to integrate this? thanks. i've tried setting $u=(\log x)^{-1}$. $\dfrac{\mathrm du}{\mathrm dx} = x^{-1}$ But it didnt ...
0
votes
1answer
74 views

About integration on manifold and partition of unity (and finiteness of open covers)

Please see the definition below of integration over a boundary of a Lipschitz domain. My question is, the summation in (C.36) for example is over $n$. But when is this a finite sum? If ...
2
votes
1answer
110 views

Evaluation of a definite integral

I want to find the best way to show $\int_0^\infty\dfrac{x^{2m}}{x^{2n}+1}\,dx=\dfrac{\pi}{2n}\operatorname{csc}\left(\dfrac{2m+1}{2n}\pi\right)$, where $0\leq m<n$. It's easy to verify some ...
2
votes
1answer
114 views

Integration, Lebesgue and counting measure

Could you help me with the following exercise? Consider $X=Y=[0,1]$ with Lebesgue measure $m$ on $X$ and counting measure $\omega$ on $Y$. Let $f:X \times Y \rightarrow \mathbb{R}$ and $f(x,y)= ...
1
vote
3answers
57 views

Function two variables, integral

Could you explain to me how to solve this exercise? $$f\colon \mathbb{R}^2 \ \rightarrow \mathbb{R}$$ $$f(x,y) = \begin{cases} 1, \ \ \ \ \ \text{for} \ x \ge 0, \ x \le y < x+1 \\ -1, \ \ ...
10
votes
0answers
411 views

Prove $\int_{0}^{\pi}\frac{x^2}{\sqrt{5}-2\cos{x}}dx=\frac{\pi^3}{15}+2\pi\ln^2{\left(\frac{1+\sqrt{5}}{2}\right)}$ without contour integration [duplicate]

Show that $$\int_{0}^{\pi}\dfrac{x^2}{\sqrt{5}-2\cos{x}}dx=\dfrac{\pi^3}{15}+2\pi\ln^2{\left(\dfrac{1+\sqrt{5}}{2}\right)}$$ This thread demonstrates how contour integration can be used to ...
3
votes
4answers
317 views

How to find the integral $\displaystyle\int_{0}^{\infty}\frac{\arctan{x}}{2+x^2}dx$

Find this integral $$I=\int_{0}^{\infty}\dfrac{\arctan{x}}{2+x^2}dx$$ My try: since let$$\arctan{x}=u$$ then $$I=\int_{0}^{\frac{\pi}{2}}\dfrac{x}{1+\cos^2{x}}dx$$ the wolf <--- then I ...
0
votes
4answers
69 views

How to calculate $\iint_D y\, dx\, dy$, $(D = \{(x, y) : x \le y \le 2x, 0 \le x \le 3\} )$?

How to calculate $$\iint_D y\, dx\, dy,$$ where $$D = \{(x, y) \mid x \le y \le 2x, 0 \le x \le 3\} )?$$ This is the first time I have to solve such question, so I have definitely no idea how to do ...
2
votes
0answers
31 views

How to calculate the following 3D ${\bf k}$-space integral?

I'm struggling to calculate$$ \sum_{a,b=\pm}\int\frac{\text d\mathbf{k}}{(2\pi)^3} ...
4
votes
0answers
123 views

How to integrate $\left(1+\ln(x)\right)\sqrt{1+(x\ln(x))^2}$ with Risch algorithm?

How would you integrate $\left(1 + \ln\left(x\right)\right)\, \sqrt{1 + \left(x\ln\left(x\right)\right)^{2}\,}$ using the Risch algorithm? I want to know this because Mathematica is using the Risch ...
1
vote
1answer
39 views

Proving a claim $|c_n e^{in\theta}| = |c_n|$

I'm studying about Fourier series from a book called "Fourier series and its applications" by Folland and on page 40, the author makes the claim that: $$|c_n e^{in\theta}| = |c_n|,$$ where $n$ is an ...
1
vote
3answers
218 views

Possible values of $\int \frac{dz}{\sqrt{1-z^2}}$ over a closed curve in a region?

This is related to Ahlfors' problem #5 following section 4.4.7. Let $\sigma$ be a path in $\mathbb{C}$ starting at $-1$ and ending at $+1$. Let $\gamma$ be a closed curve in $\mathbb{C}$ which does ...
0
votes
1answer
61 views

Cannot solve indefinite integral

can You help me with this indefinite integral $$\int \dfrac{2^{x + 1} + 5^x}{10^x} dx$$ What to use. direct integration or substitute of variables Thanks
0
votes
2answers
54 views

Help me with indefinite integral

Can You help me with it $$\int\ \frac{tg^2 x}{ \cos^2x}\ dx$$ I've already tried to decompose tg. But I don't know what to do after that.
0
votes
1answer
29 views

Calculate the line integral

Calculate $\int_{|z|=1}{\frac{|dz|}{|z-c|^2}}$ when $|c|<1$. I've tried to substitute $z$ for $e^{i\theta}$ and somehow use the fact that on the unit circle $\overline{z}=\frac{1}{z}$ but I have ...
1
vote
0answers
37 views

Line integral, Parametrization

I have this line $A=\{(x,y) \in R^2 : y^2+4x^4-4x^2=0\}$ , $(x>0)$ I parametrized it like that : $b(t) = (t, \sqrt{4t^2- 4t^4})$. And my $F$ is $F(x,y) = (x+y,-x)$. But when I calculate my ...
0
votes
0answers
47 views

How to find an error estimate for integral of curvilinear surface triangle when using quadrature

I would like to find a way to estimate the error due to the calculation of the normal when one tries to find the volume of a volume composed of quadratic surface triangles using numerical gauss ...
1
vote
2answers
42 views

Calculate the value of c for which f is a probability density.

Let f the function defined by: Where c is positive none zero and constant . How can i calculate the value of c for which f is a probability density.Thnxs for the help.
1
vote
1answer
60 views

How to calculate gaussian integal

we know that gaussian kernel is defined by $K_\sigma(x-y)=\exp\frac{-\|x-y\|^2}{2\sigma^2}$ I want to calculate integral of this function: $$H= \int K_\sigma(x−y)\cdot f(y)dx=f(y)\int ...
12
votes
3answers
393 views

Evaluate $\int_0^{\frac{\pi}{2}}\frac{x^2}{1+\cos^2 x}dx$

Evaluate the following integral $$\int_0^{\frac{\pi}{2}}\frac{x^2}{1+\cos^2 x}dx$$ This function does not have an elementary anti-derivative. How can we solve this?
1
vote
2answers
85 views

calculate an integration by using residue theorem

$$\int_0^\infty\frac1{(x^2+1)(\ln^2(x)+\pi^2)}$$ Calculate the following integration using the residue theorem:
2
votes
2answers
208 views

Integral $\int_{0}^{1}\frac{1}{x^{2}+2x+2}dx$ via contour integration

I want to evaluate the following integral $$\int_{0}^{1}\frac{1}{x^{2}+2x+2}dx$$ by contour integration; I have a problem with the choice of the contour/ branch cuts. Where can I find some some ...