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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0
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4answers
81 views

Integration $1/x$ - complex number

Why there is no integral $$\int_{-e}^{e}\frac{1}{x}$$ And why integral $$\int_{-e}^{-1}\frac{1}{x}= -1$$ and not $$\int_{-e}^{-1}\frac{1}{x}=(-1 + i\cdot\pi)$$ E.g. ...
0
votes
1answer
58 views

Change of variables - integrals

\begin{equation} \text{Let $\hspace{3mm}$ }f(t) = 2\int_{b}^{\infty} \sqrt{\frac{1}{2\pi t}}e^{-x^2/2t}dx. \end{equation} I found that this integral can be written with change of variables can be ...
0
votes
1answer
219 views

integrating: $\sqrt{1+4x^{2}+4x^{4}}$ on the interval [0,1]

so far i've tried to use substitution with $u = x^{2}$ but this didn't make my integral easier to calculate. i don't see how i can use partial integration either so i get kind of stuck. Wolfram alpha ...
5
votes
2answers
194 views

How to prove $\int_0^\infty e^{-x}\frac{\sin^2 x}{x}\text{ dx}=\frac{\text{log }5}{4}$

I'm asked to prove $$\displaystyle\int_0^\infty e^{-x}\frac{\sin^2 x}{x}\text{ dx}=\frac{\text{log }5}{4}\tag{$\ast$}$$ by integration of $e^{-x}\text{sin}(2xy)$ over an suitable measurable ...
8
votes
6answers
2k views

How to integrate a three products

I tried to integrate $x e^x \sin x$, using integration by parts, and setting $dv/dx = e^x \sin x$. Even though I got really close, I kept getting it wrong. Can someone please solve it with working ...
2
votes
0answers
56 views

Formula for integral over hypersurface??

Can someone give me a formula for an (Lebesgue) integral of a function $f:M \to \mathbb{R}$ where $M$ is a bounded $C^k$-hypersurface of dimension $(n-1)$ in $\mathbb{R}^n$? I have tried the ...
0
votes
2answers
141 views

Flux through cone and hyperboloid

I want to find the flux of the vector field $$F(x,y,z)=(x+z,2x+2y,3y+3z)$$ through the body defined by $$x^2/16+y^2/4-z^2/3\le 1 \quad\mbox{with} \quad 0\le z \lt 3$$ $$x^2/16+y^2/4-(z-7)^2/4\le 0 ...
4
votes
1answer
68 views

Find recursive forumula for integrals

I have to find recursve formulas for solving the following two integrals. The assignment tells one to find an Expression that leads from the calculation of $\dfrac{I_{2n}}{I_{2n+1}}$ to the ...
0
votes
1answer
20 views

Given $\int\int_D \arctan \frac{y}{x}dxdy $ where $D = \{(x, y):1 \le x^2 + y^2 \le 4, x \le y \le \sqrt3x, x \ge 0 \}$. Move to polar coordinates?

Given $\int\int_D \arctan \frac{y}{x}dxdy $ where $D = \{(x, y):1 \le x^2 + y^2 \le 4, x \le y \le \sqrt3x, x \ge 0 \}$. Move to polar coordinates. I stuck with finding $\theta$. I know that $r ...
2
votes
1answer
53 views

Integration on compact manifold

Integration on a nice enough manifold of a function $f:M \to \mathbb{R}$ is defined $$\int f = \sum_{ i \in I} \int_{U_i}\phi_i f$$ where $\phi_i$ is a partition of unity subordinate to the open cover ...
1
vote
1answer
807 views

Midpoint approximation over/under estimation

So left handed approximation underestimates the area under a increasing curve and over estimates for decreasing curves. And right handed approximation overestimates for increasing curves and ...
2
votes
0answers
117 views

What is an example of an integral that CANNOT be done without contour integration ? If that exist.

What is an example of an indefinite integral that CANNOT be done without contour integration ? If that exist. Im talking about closed forms for integrals, not numerical methods. Note that there are ...
-1
votes
2answers
81 views

Integrate by parts $\int x^3e^{x^2}dx$

Integrate by parts: $$\int x^3e^{x^2}dx$$ I see a way by substitution, but I'm not seeing the integration by parts. Taking $u=e^{x^2}$ seems not to work, and I believe $e^{x^2}$ is not ...
0
votes
2answers
52 views

How can I evaluate $\int \frac{5x^3+2}{x^3-5x^2+4x}dx$

How can I calculate the following: $$\int \frac{5x^3+2}{x^3-5x^2+4x}dx$$ I think long division might help me, but I can't understand how to do it here, nor if it will help. Thanks a lot !
1
vote
1answer
32 views

Finding the parent function of the integral.

How to deal with this integral? $\int{ \frac{dx}{(x^{2}+2x+10)^{3}}}$
0
votes
3answers
69 views

minimal value of integral

I have an integral $$\int_0^{\pi/2}\left(\cos(x)-ax\right)^2\,dx$$ and I need to know for which a the function is minimal. so I can take the derivative of the integral, but what should I do with ...
2
votes
3answers
132 views

Asymptotic behavior of an integral

I am interested in the integral \begin{align*} ...
4
votes
2answers
117 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
158 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
122 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 ...
2
votes
1answer
290 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 ...
1
vote
2answers
61 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
39 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
224 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
438 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
112 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
211 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
57 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
63 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
160 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
145 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
42 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, ..., ...
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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
87 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
161 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
274 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
130 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
78 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
111 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
128 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
413 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 ...