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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2
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1answer
83 views

Evaluate $\iint\limits_{\substack{x<u,y<v, \\ x^2+y^2<1}} dxdy$

How can I evaluate the following double integral: $$\iint\limits_{\substack{x<u,y<v, \\ x^2+y^2<1}} dxdy$$ If we didn't have the restrictions $x<u, y<v$ polar coordinates would have ...
1
vote
3answers
69 views

Definite Integration problem

I was working on a small Integration problem, where i was needed to solve another integral: $$I= \int_0^{\pi} \frac{1}{1+\cos^2x} dx $$ Working: $$\int_0^{\pi} \frac{1}{1+\cos^2x} dx=\int_0^{\pi} ...
0
votes
1answer
43 views

Finding the types of singularities of $\oint \frac{\sin(\pi \cdot z)}{(z-1)^2}$

I want to find the types of singularities of $$\oint \frac{\sin(\pi \cdot z)}{(z-1)^2}$$ the point is $z=1$ I know that: $$f(z)=\frac{p(z)}{q(z)},q(a)=0,p(a)\neq 0,p(z)$$ so $p(z)$ analytic in $a$ ...
2
votes
1answer
148 views

How to show that the difference of two Gumbel distributed random variables follows a Logistic distribution?

How can you show that when you have two random variables $X,Y\sim\text{Gumbel}[0,1]$ , then $X-Y\sim\text{Logistic}[0,1]$ . I tried to use the convolution formula ...
0
votes
2answers
77 views

Can't read integral method [closed]

I type this : fun = @(x) exp(-x.^2).*log(x).^2; q = integral(fun,2,4); q; when I run the above code, I get an error message Undefined function or method ...
0
votes
4answers
200 views

how to evaluate $\int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin x}}\text{d}x$

I was solving a physics problem and eventually the problem boiled down to solving the following integral: $$\int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin x}}\text{d}x$$ I have already tried ...
1
vote
1answer
75 views

One can integrate every monotonic function

I have a question related to the proof of "One can integrate every monotonic fucktion $f: [a,b] \to \mathbb R$." that I have as assignment. We are referring to Riemann integrals here. The idea I came ...
2
votes
5answers
105 views

I am going to learn these Mathematics Topics. I need advice and suggestions please .

I am really horrible when it comes to maths since I never had any maths background in my High school. I am fairly good at programming ( C++ and Java) but without mathematics I cant advance in any ...
2
votes
4answers
160 views

How to integrate $\int_{0}^{\pi /2} e^{−2x}\sin(3x)\rm dx $?

How to integrate $$\int_{0}^{\pi/2} e^{−2x}\sin(3x)\rm dx $$ I have attempted to this question with integration by parts, but I'm hitting a lot of walls. I have a feeling it might have to do with ...
6
votes
2answers
71 views

Find $\int_0^{+\infty}\cos 2x\prod_{n=1}^{\infty}\cos\frac{x}{n}dx$

Evaluate the following integral $$\int_0^{+\infty}\cos 2x\prod_{n=1}^{\infty}\cos\frac{x}{n}dx$$ I was thinking of a way which do not need to explicitly find the closed form of the infinite product, ...
0
votes
1answer
73 views

Real valued function generalizes to functions on $f:[a,b] \to V$

Let $[a,b]$ be a closed interval in $\mathbb{R}$ and let $V$ be a complete normed vector space. Prove that if $f:[a,b]\to V$ is continuous then $||\int^b_af(x)d(x)||\le \int^b_a||f(x)||d(x).$ ...
0
votes
3answers
143 views

Is this integral true? Or is it too much for Wolfram Alpha?

I was playing around with the wolfram calculator, just adding different things and mesmorising at what they equaled, then I randomly put in a bunch of trig functions, and well, this is what I got: $$ ...
0
votes
1answer
63 views

Could someone explain this(integral definition from Spivak's Calculus)

He gives a function $f(x) = 0, x \neq 1$ and $f(x) =1, x =1$ .Basically the function gives 1 of it's 1 and 0 if it's anything else.This is the function graph: Then we suppose $P = ...
2
votes
5answers
150 views

What's the simplest way to solve $\int \frac{1}{4-v^2}dv$?

I did a substitution on a DE and ended up with this: $\int \frac{1}{4-v^2}dv$ I tried a trig substitute but things got a bit hairy. WolframAlpha recommended a far less intuitive substitution. I ...
2
votes
2answers
76 views

Derivative of area of ellipse with respect to axis

Suppose $A(a,b)$ defines the area of an ellipse with axes $a,b$. We know that $A(a,b)=\pi ab$, and so $\partial_bA(a,b) = \pi a$.  But suppose I parameterize the ellipse in polar coordinates as ...
0
votes
2answers
30 views

Integrating over a region where a function of two variables is less than some value

Say I have $$ f(x,y)=e^{b(x^2+y^2)} $$ and I want to integrate over the region where $f(x,y)<a$. What is the best way to work out the limits on the integral (one of them being a function of $x$ ...
3
votes
0answers
68 views

Determine the behavior of a function defined by an integral

Suppose we have a function defined by $$\varphi(s)=\int_{-\infty}^\infty f(x,s)\,dx$$ defined for $s\in S\subseteq \mathbb{R}$. Suppose we know that it blows up at $a\in \partial S$, and we want to ...
1
vote
1answer
83 views

Change the integration order of $\int^2_{-6} \int^{2-x}_{\frac{x^2}{4}- 1}f(x, y) \, dy\,dx$

This is a question from sample exam that I'm trying to solve but having difficults. Change the integration order of the integral: $$\int^2_{-6} \int^{2-x}_{\frac{x^2}{4}- 1}f(x, y) \, dy\,dx$$. ...
0
votes
2answers
73 views

Integral of a Kernel function

I try to understand p.334 question 9-1 in Cameron Trivedi (link) where I have to calculate the bias of a Kernel density estimate at x=1 and n=100, where we assume that the underlying density is ...
4
votes
2answers
111 views

Integral in 3 dimensions

I am trying to integrate $$ \iiint \delta(|\mathbf r| -R)\:\mathrm{d}^{3}\mathbf{r} $$ I know that $ \int f(r) \delta(r-R) d^3 \mathbf r =f(R) $, but when I try to apply this here I end up ...
1
vote
4answers
53 views

Calculating the are between 2 functions, by y-axis

So, I have these 2 equations: $$ y_1=e^{2x}\\y_2=-e^{2x}+4 $$ And I need to calculate the area they have limited right of the y-axis this is how it looks: I have converted the formulas to x: $$ x_1 = ...
0
votes
3answers
108 views

Integrating $\int\sin^{-2}xdx$ [duplicate]

I am trying to prove that $$ \int\frac{1}{\sin^2(x)}dx = -\cot(x) + C $$ but I have difficulties, I don't know where to start, I can't substitute anything with $sin(x)$ because I don't have a $cos(x)$ ...
3
votes
2answers
87 views

How do we prove $\int_I\int_x^1\frac{1}{t}f(t)\text{ dt}\text{ dx}=\int_If(x)\text{ dx}$

Let $f:\mathbb{R}\to\mathbb{R}$ be Borel-measurable and Lebesgue-integrable over $I:=(0,1)$. Further, let $\;\;\;\;\;\;\;\;\;\;g : I\to \mathbb{R}\;,\;\;\; \displaystyle x ...
0
votes
1answer
34 views

Change of variable with measures other than the Lebesgue measure.

I ask my question with a specific example in mind. Consider the integral \begin{align} I_k=\int_{\mathbb R}(2\cos(x))^k~d\mu(x),&&k\in\mathbb N\tag{1} \end{align} where ...
0
votes
2answers
173 views

What is the integration of integral of $\cot(\log(\sin(x))$ ??

what is the integration of integral $$\int\cot(\log(\sin(x))\, dx$$ I have tried: Let $\log(\sin(x))=z$ or $\dfrac{1}{\sin(x)}\cos(x)\cdot dx=dz$ , that means $\cot(x)\cdot dx=dz$ $\Longrightarrow$ ...
0
votes
4answers
80 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
57 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
192 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
192 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 ...
7
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
51 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
133 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
67 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
669 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
113 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
51 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
30 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
68 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
115 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
153 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
121 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
266 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
60 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 ...